Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Notes on Spinner Learning Task 3 - Georgia Department of Education
Document related concepts
Transcript
Mathematics I Frameworks Teacher’s Edition Unit 4 The Chance of Winning 1st Edition March 21, 2008 Georgia Department of Education Mathematics I Unit 4 1st Edition Table of Contents NAEP Sample Items..................................................................................................................................... 3 Introduction: ................................................................................................................................................ 5 Notes on Spinner Learning Tasks ............................................................................................................. 12 Notes on Spinner Learning Task 1............................................................................................................ 13 Notes on Spinner Learning Task 2............................................................................................................ 16 Notes on Spinner Learning Task 3............................................................................................................ 18 Notes on Spinner Learning Task 4............................................................................................................ 21 Notes on Spinner Learning Task 5: .......................................................................................................... 24 Notes on Spinner Learning Task 6............................................................................................................ 27 Notes on Spinner Learning Task 7............................................................................................................ 28 Notes on Spinner Learning Task 8............................................................................................................ 29 Notes on Spinner Learning Task 9............................................................................................................ 32 Notes on Testing Learning Task 1............................................................................................................. 33 Notes on Testing Learning Task 2: ........................................................................................................... 37 Notes on Testing Learning Task 3............................................................................................................. 39 Notes on Testing Learning Task 4............................................................................................................. 41 Notes on Testing Learning Task 5............................................................................................................. 42 Notes on Survey Learning Task ................................................................................................................ 44 Notes on Medical Learning Task .............................................................................................................. 45 Notes on Area Learning Task .................................................................................................................... 46 Notes on Card Learning Task.................................................................................................................... 47 Notes on Marble Learning Task ................................................................................................................ 48 Notes on Dice Learning Task .................................................................................................................... 50 Notes on Simulation Learning Task.......................................................................................................... 53 Notes on Game 1 Learning Task ............................................................................................................... 55 Notes on Game 2 Learning Task ............................................................................................................... 56 Notes on Game 3 Learning Task ............................................................................................................... 58 Notes on Culminating Task ....................................................................................................................... 59 Additional Tasks that can be used as Culminating Tasks ........................................................................ 62 Sample Homework .................................................................................................................................... 68 Solutions ..................................................................................................................................................... 87 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 2 of 120 Mathematics I Unit 4 1st Edition Unit 4: Sample Assessment Questions from NAEP 1. The two fair spinners shown to the right are part of a carnival game. A player wins a prize only when both arrows land on black after each spinner has been spun once. James thinks he has a 50-50 chance of winning. Do you agree? 2. A certain company keeps a list of 50 employees and their annual salaries. When the salary of the very highly paid president is added to this list, which of the following statistics is most likely to be approximately the same or nearly the same for the original list and the new list? A) The B) The C) The D) The highest salary range mean median E) The standard deviation 3. This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all of your work. The table below shows the daily attendance at two movie theaters for 5 days and the mean(average) and the median attendance. Theater A Theater B Day 1 100 72 Day 2 87 97 Day 3 90 70 Day 4 10 71 Day 5 91 100 Mean (average) 75.6 82 Median 90 72 (a) Which statistic, the mean or the median, would you use to describe the typical daily attendance for the 5 days at Theater A? Justify your answer. (b) Which statistic, the mean or the median, would you use to describe the typical daily attendance for the 5 days at Theater B? Justify your answer. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 3 of 120 Mathematics I Unit 4 1st Edition 4. Four people - A, X, Y, and Z - go to a movie and sit in adjacent seats. If A sits in the aisle seat, list all possible arrangements of the other three people. One of the arrangements is shown below. 5. From a shipment of 500 batteries, a sample of 25 was selected at random and tested. If 2 batteries in the sample were found to be dead, how many dead batteries would be expected in the entire shipment? A) 10 B) 20 C) 30 D) 40 E) 50 6. The pulse rate for a group of 100 people is shown in the graph. What is the average pulse rate per minute for these 100 people? (Note: Use the midpoint of each interval to represent the pulse rate for the entire interval. For example, 55 would be used for the pulse rate of the 15 people in the 50-60 group.) 7. In the graph above, each dot shows the number of sit-ups and the corresponding age for one of 13 people. According to this graph, what is the median number of sit-ups for these 13 people? A) 15 B) 20 C) 45 D) 50 E) 55 8. A fair coin is to be tossed three times. What is the probability that 2 heads and 1 tail in any order will come up? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 4 of 120 Mathematics I Unit 4 1st Edition Mathematics 1 - Unit 4 The Chance of Winning Teacher’s Edition Introduction: In this unit, students will calculate probabilities based on angles and area models, compute simple permutations and combinations, calculate and display summary statistics, and calculate expected values. They should also be able to use simulations and statistics as tools to answering difficult theoretical probability questions. Enduring Understandings: Using mathematical skills acquired from statistics and probability, students can better determine whether games of chance are really fair. Students should be able to use mathematics to improve their strategies in games. Key Standards Addressed: MM1D1 Students will determine the number of outcomes related to a given event. a. Apply the addition and multiplication principles of counting b. Calculate and use simple permutations and combinations MM1D2. Students will use the basic laws of probabilities. a. Find the probabilities of mutually exclusive events b. Find probabilities of dependent events c. Calculate conditional probabilities d. Use expected value to predict outcomes MM1D3. Students will relate samples to a population. a. Compare summary statistics (mean, median, quartiles, and interquartile range) from one sample data distribution to another sample data distribution in describing center and variability of the data distributions. b. Compare the averages of summary statistics from a large number of samples to the corresponding population parameters c. Understand that a random sample is used to improve the chance of selecting a representative sample. MM1D4. Students will explore variability of data by determining the mean absolute deviation (the averages of the absolute values of the deviations). RELATED STANDARDS ADDRESSED: MM1G2. Students will understand and use the language of mathematical argument and justification. a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate MM1P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 5 of 120 Mathematics I Unit 4 1st Edition b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. MM1P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. MM1P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. MM1P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. MM1P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. UNIT OVERVIEW Students should already have knowledge that probabilities range from 0 to 1 inclusive. They should also be able to determine the probability of an event given a sample space. They should be able to calculate the areas of geometrical figures and measure an angle with a protractor. Sometimes when studying probability, it is easier to understand how to find an answer by examining a smaller sample space. The wheel used on Wheel of Fortune has many different sections. It also has “lose a turn” and “bankrupt” which turns a simple probability problem into one that is much more complex. In addition, each section of the wheel may not have the same area; therefore, this type of spinner may be different from the ones that are familiar to students. Permutations versus Combinations: Students tend to confuse permutations with combinations. When teaching this portion of the unit, I would suggest integrating permutations with combinations with simple problems involving the multiplication principle. Students need many opportunities to decide which Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 6 of 120 Mathematics I Unit 4 1st Edition formula to use in which context prior to a unit assessment. You also may want to refrain from giving them the formula for permutations and combinations immediately. Instead, students should discover the patterns first before they see the formula. This should make the formulas more meaningful and help with retention. When the sample space is too large to be represented by a tree diagram: It’s easy to write the sample space for flipping a coin 4 times and determining the probability that you have at least 2 heads. However, problems arise when you ask them to find the probability of at least 2 heads when flipping the coin 20 times since the sample size is very large 2 20 = 1048576. (Try listing those outcomes in a 50 minute class period!) To solve this problem, you may have students explore patterns in smaller sample spaces. Have students draw the tree diagrams for 2 flips, 4 flips, 6 flips, etc. Ask students to examine the patterns in the sizes of the sample spaces to help them determine the size of the sample space for 20 flips. Have them find a strategy, based on these smaller sample sizes, to come up with a way to count “at least 2 heads for 20 flips.” When events are not equally likely: In middle school, students may have only used tree diagrams for equally likely events (flipping a fair coin, rolling a fair die, etc.). If the events are equally likely, the branches of the tree diagram do not have to be labeled with the associated probabilities for students to get the correct probability. For example, suppose a fair coin is tossed twice. If a tree diagram is used to determine the probability of getting a “head” on the first flip and a “tail” on the second flip, students can easily see the sample space, ( H , H ),(T , T ),( H , T ),(T , H ) , and realize that HT occurs once out of 4 times. Students can use the multiplication principle to confirm that the probability of HT is (.5)(.5)=.25. Thus, it would not matter whether the students labeled the branches of the tree diagram with the associated probabilities if the coin is fair. Suppose that the coin is not fair. Suppose the probability of heads is .6 and the probability of tails is .4. Then the probability of HT = (.6)(.4) = .24 not .25. The sample space is still (H , H ),(T , T ),(H , T ),(T , H ) , but the probability of HT is no longer ¼ since the probability of heads is not the same as the probability of tails. To possibly avoid this problem, ask the students to label the branches tree diagram with their associated probabilities. When students cannot calculate the probability of an event: If students don’t understand the theory, use simulations! Many adults as well as students struggle with probability. A good example of this is the classic “Monty’s Dilemma” problem addressed in the “Ask Marilyn” column. The question was based on the popular “Let’s Make a Deal” show. At one point on the show, there were 3 curtains. Behind one curtain was a great prize. Behind the other two curtains were awful prizes. The show’s host, Monty, asked the player to pick a curtain. After the player picked a curtain, the host revealed a prize behind one of the other two curtains (not the good prize). The host then asked the player if he would like to stay or switch. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 7 of 120 Mathematics I Unit 4 1st Edition Marilyn stated that the player should switch because the probability that he picked the grand prize from the beginning was 1/3. So, the probability of winning would be 2/3 if the player switched. She received many letters, some from mathematicians, claiming that it would not matter if the player stayed or switched…the probability of winning now changed to 1/2 since one bad prize was revealed. She then asked middle schools across the country to simulate this and send her the results. The results stated that it was better to switch. Many times, “real life” probability is difficult to compute or hard to understand. That is why it’s so important to be able to perform simulations. With computers and graphing calculators readily available, simulations are easy to perform and are not time consuming. Understanding conditional probability: Although there are other methods, I typically teach my students the following two techniques to solve conditional probability problems. Technique #1: Think of the sample space described. For example, suppose the question reads, “Given a person rolls an even number on a die, what is the probability that the die lands on a 2?” I ask students to list the sample space described. It is not all possible outcomes on the die because we know that the person rolled an even number. The sample space is just 2, 4,6 . Therefore, the probability is 1/3. Technique #2: Use a tree diagram and the conditional probability formula. The formula for conditional probability is P(A given B) = P(A and B)/P(B). It’s symbolically written as such P( A B) where stands for the intersection of sets A and B. P( A B) P( B) This formula for the previous problem would be much more difficult than technique #1. Using the formula, the numerator would be 1 P( A B) P(rolls a 2 and rolls an even number ) . 6 3 The denominator would be P( B) P(even number ) . 6 1 1 We would get the same result, 6 but it would be a little more difficult and time 3 3 6 consuming for students. The formula, however, does have its merit. Sometimes, it is not easy or possible to list the sample space. In that case, the formula is necessary. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 8 of 120 Mathematics I 1st Edition Unit 4 A problem which requires the formula might read, “Suppose a student knows 30% of the class material without studying for an upcoming multiple choice test which has 15 questions (4 possible answers per question). Suppose the student does not study for the test. If she provides the correct answer on the test, what is the probability that she strictly guessed?” I have found tree diagrams very helpful to solve these types of conditional probability problems. Students should already be familiar with constructing tree diagrams from middle school. Formulas and Definitions Addition Rule for mutually exclusive (disjoint) events: P( A or B ) P ( A) P ( B ) Addition Rule for sets that are not mutually exclusive: P( A or B) P( A) P( B) P( A B) Census: A census occurs when everyone in the population is contacted. Conditional Probability: P( A B) Combinations: Complement: This refers to the probability of the event not occurring P( Ac ) 1 P( A) Dependent: Two events are dependent when the outcome of the first event affects the probability of the second event. For example, suppose two cards are drawn from a standard deck of 52 cards without replacement. If you want the probability that both 4 3 cards are kings, then it would be . If a king was drawn first, then there would 52 51 only be 3 kings left out of 51 cards since the first king was not put back in the deck. Hence, the probability of drawing a king on the second draw is different than the probability of drawing a king on the first draw, and the events are dependent. Expected Value: The mean of a random variable X is called the expected value of X. It n Cr P( A B) P( B) n! r !(n r )! n can be found with the formula X P where P i 1 i i i is the probability of the value of X i . For example: if you and three friends each contribute $3 for a total of $12 to be spent by the one whose name is randomly drawn, then one of the four gets the $12 and three of the four gets $0. Since everyone contributed $3, one gains $9 and the other three looses $3. Then the expected value for each member of the group is found by (.25)(12) +(.75)(0) = 3. That is to say that each pays in the $3 expecting to get $3 in return. However, one person gets $12 and the rest get $0. A game or situation in which the expected value is equal to the cost (no net gain nor loss) is commonly called a "fair game." However, if Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 9 of 120 Mathematics I Unit 4 1st Edition you are allowed to put your name into the drawing twice, the expected value is (.20)(12)+(.80)(0) = $2.40. That is to say that each pays in the $3 expecting to get $2.40 (indicating a loss of $.60) in return. This game is not fair. Fair: In lay terms it is thought of as "getting the outcome one would expect," not as all outcomes are equally likely. If a coin (which has two outcomes...heads or tails) is fair, then the probability of heads = probability of tails = 1/2. If a spinner is divided into two sections...one with a central angle of 120 degrees and the other with an angle of 240 degrees, then it would be fair if the probability of landing on the first section is 120/360 or 1/3 and the probability of landing on the second section is 240/360 or 2/3. Independent: Two events are independent if the outcome of the first event does not affect the probability of the second event. For example, outcomes from rolling a fair die can be 1 considered independent. The probability that you roll a 2 the first time is . If you roll 6 the die again, there are still 6 outcomes, so the probability that you roll a 2 the second 1 time is still . 6 Measures of Center n Mean: The average = X i 1 i . The symbol for the sample mean is X . The N symbol for the population mean is X . Median: When the data points are organized from least to greatest, the median is the middle number. If there is an even number of data points, the median is the average of the two middle numbers. Mode: The most frequent value in the data set. Measures of Spread (or variability) Interquartile Range: Q3 Q1 where Q3 is the 75th percentile (or the median of the second half of the data set) and Q1 is the 25th percentile (or the median of the first half of the data set). X i X where X is each individual data point, X is the Mean Deviation: i N sample mean, and N is the sample size Multiplication Rule for Independent events: P( A and B) P( A) P ( B ) Mutually Exclusive: Two events are mutually exclusive (or disjoint) if they have no outcomes in common. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 10 of 120 Mathematics I Unit 4 1st Edition Parameters: These are numerical values that describe the population. The population mean is symbolically represented by the parameter X . The population standard deviation is symbolically represented by the parameter X . n! Permutations: n P r (n r )! Random: Events are random when individual outcomes are uncertain. However, there is a regular distribution of outcomes in a large number of repetitions. For example, if you flip a fair coin 1000 times, you will probably get tails about 500 times. But, you probably won't get HTHTHTHT or even HTHTHHTTHHTT when you flip the coin, so the outcome is uncertain for each flip. Or, if you roll two dice and record the sums 1000 times, you will probably get about 167 sums of 7, 139 sums of 6, etc. which are the expected values ( expected value for a sum of 7 is 6/36*1000 =167). Hence, we will have a regular distribution of outcomes. However, since rolling two fair dice is a random event, we won't know what sum our dice will give on each roll. Sample: A subset, or portion, of the population. Sample Space: The set of all possible outcomes. Statistics: These are numerical values that describe the sample. The sample mean is symbolically represented by the statistic X . The sample standard deviation is symbolically represented by the statistic sx . TASKS: The remaining content of this framework consists of student tasks. The first is intended to launch the unit by developing a basic understanding of what random and fair are. Tasks are designed to allow students to build their own understanding through exploration follow. The tasks fall under three basic considerations: Wheel of Fortune, True or False, and Yahtzee. The students will find these tasks engaging, mathematically rich and rigorous. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 11 of 120 Mathematics I Unit 4 1st Edition Notes on Spinner Learning Tasks Comments: Show the students a five minute clip of the show, Wheel of Fortune, if you have permission. Or, describe the show to them. Ask the students questions that you would think would be interesting to explore. The questions that will be discussed in this part of the unit are: 1) Is it just as likely to land on bankrupt as the big money? Is the spinner really random? What does random mean? 2) If I think that I know the phrase, should I spin to accumulate more money? Or, should I guess and take a chance of losing my turn or going bankrupt? 3) If there are only 3 letters left to guess and I spin the wheel 3 times in a row, how much money (on average) would I expect to accumulate? 4) If I don’t know the phrase, which letters should I guess? Tell them that they will explore many of those questions with this unit. But, in order to explore complex probabilities, it is easier to start with much simpler ones. So, for the next couple of days, the spinners used will not be as nearly complex as the one on the game show. But, by the end of the unit, they should be able to understand more about the spinners on the game show. Materials needed: The students should have protractors to measure angles. You may want to provide them with the spinners. Look under “Spinner Master” if you would like to use the spinners in this unit. You will need brads and either paper clips or safety pins for the arrows. You can copy the spinners to cardstock. Also, you will need post-it notes for the bar graph activity. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 12 of 120 Mathematics I 1st Edition Unit 4 Notes on Spinner Learning Task 1 $300 $600 $500 $100 $800 $700 $200 $400 Make an arrow for the spinner above so that the spinner is useable. You can use brads and a paper clip for an arrow. Is this homemade spinner fair? How can you tell? Comments: Discuss the spinners with the students. Ask: How can we tell if spinning a spinner is really a random event? Do you think that your homemade spinner is just as fair as a store bought spinner? How can you tell? You may need to encourage students to perform a simulation to determine whether their spinner is fair. Each student should spin his/her spinner at least 50 times and record the outcomes on his/her paper. Each student can make a dot plot of his simulation result. Ask the students, “If the spinner is fair, what should be the shape of the dot plot? Should it appear skewed to the right (most of the data on the left with fewer data on the right)? Should it appear skewed left? Should it appear bell-shaped (symmetrical graph with most of the data in the center), or should it appear uniform in height (each bar the same in height)? You may need to discuss the definition of a random event with students. Events are random when individual outcomes are uncertain. However, there is a regular distribution of outcomes in a large number of repetitions. Discuss whether or not their personal spinners are random. You may want to pool the class data to see if a paper clip or safety pin weights the spinner in such a way that it is not fair. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 13 of 120 Mathematics I 1st Edition Unit 4 Calculate the following probabilities for the spinner (assuming the spinner is fair). Comments: Assuming that their spinner is working properly and is fair, ask the students the following questions. I would think that many students can already answer these. You can formally introduce the notion of mutually exclusive events along with its definitions when you get to problem #4 (MM1D2a) $300 $600 $500 $100 $800 1) What is the probability of obtaining $800 on the first spin? $700 $200 1/8 $400 2) What is the probability of obtaining $400 on the first spin? 1/8 3) Is it just as likely to land on $100 as it is on $800? Yes since the probabilities of landing on $100 or $800 are the same. 4) What is the probability of obtaining at least $500 on the first spin? ½ (at least $500 means $500 or $600 or $700 or $800) 5) What is the probability of obtaining less than $200 on the first spin? 1/8 (less than $200 means only $100) 6) What is the probability of obtaining at most $500 on the first spin? 5/8 (at most $500 means $100 or $200 or $300 or $400 or $500) 7) If you spin the spinner twice, what is the probability that you will have a sum of $200? 1/64 (since the only way to do this is to land on $100 then $100 again…there are 8*8=64 possible outcomes in two spins…a tree diagram would illustrate this well) 8) If you spin the spinner twice, what is the probability that you will have a sum of at most $400? 6/64 = 3/32 (300 then 100, 200 then 100, 200 then 200, 100 then 100, 100 then 200, 100, then 300….so 6/64 possibilities) Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 14 of 120 Mathematics I Unit 4 1st Edition Comments: The following questions will be a little more difficult because there are two spins involved. You can address the multiplication principle of counting here (MM1D1 a). You would probably let the students make either a tree diagram or a list of all the possibilities before you tell them that there are (8*8=64) possible sums on two spins. They may be able to pick up on that through patterns. The tree diagram is a great visual way of understanding that there are (8*8) possibilities as well as the idea that (100,200) is different outcome than (200, 100) even though it has the same sum. They should also be familiar with tree diagrams from the middle school curriculum. You may want to write the associated probabilities on each branch of the tree diagram. Since each spin is assumed to be independent of each other (for example, it is just as likely to land on $200 the second time you spin as it was the first time…the probability does not change), then you can multiply the probabilities for each spin. P(A and B) = P(A) * P(B) as long as events A and B are independent. 9) If you spin the spinner twice, what is the probability that you will have a sum of at least $1500? 3/64 (700 then 800, 800 then 800, 800 then 700….so 3/64 possibilities) 10) If you spin the spinner twice, what is the probability that you will have a sum of at least 300? 63/64 (I would use the complement to solve this problem…at least $300 is the complement of $200. The only way to get a sum of $200 is by landing on $100, $100. The probability of a sum of $200 is 1/64. So, the probability of a sum of at least $300 is 1-1/64=63/64). 11) Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200? 1/8 (You are looking at a smaller sample space. The first spin must be $100; therefore, you only have 8 possibilities which include ($100, $100), ($100, $200), ($100, $300),….($100, $800). Of those 8 possibilities, the sum is only $200 in the situation of ($100, $100).) 12) Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1000? 7/8 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 15 of 120 Mathematics I Unit 4 1st Edition Notes on Spinner Learning Task 2 Comments: Ask students: How much money, on average, would you expect to receive each time you spin the spinner? Some students may guess accurately with reasoning. If so, that is great! I would still perform the simulation to let the class see if the $300 $600 student’s guesses are correct. $500 Have each student spin the spinner 10 times, record the amounts of each spin. $100 They should then average these amounts. On your white board, the students will make a histogram with $800 their post-it notes. The x-axis will be the average value of the $700 10 spins. You could write $100-$800, but you probably will not $200 have a student with an average of $100 unless all ten spins $400 landed on $100. Each student should write the average of the 10 spins on the post it note. They will then stick the post-it note on top of the corresponding average. Once the histogram is complete, discuss how to find its center (mean or median), shape (symmetric, somewhat symmetric, or skewed), and spread (range, standard deviation, mean deviation, or interquartile range). Note: these topics will be covered extensively later. You may just want to introduce the concepts of center, shape, and spread today. Based on the histogram, let the students decide how much money they would expect to receive, on average, each time they spin the spinner. Now, you can show how to compute the theoretical expected value by hand. 1) If you spin the spinner once, how much money, on average, would you expect to receive? x p( xi ) The expected value is calculated by finding i . The probability of each section of the spinner is 1/8. So, the expected value = $100(1/8) + $200(1/8) + $300(1/8) +…. $800(1/8) = $450. 2) If you spin the spinner twice, how much money, on average, would you expect to receive? 2(450)=$900 3) If you spin the spinner 10 times, how much money, on average, would you expect to receive? 10(450)=4500. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 16 of 120 Mathematics I Unit 4 1st Edition 4) On average what is the fewest number of spins it will take to accumulate $1000 or more? To solve this, the students can simulate and use statistics to figure out the answer as seen below: Students spin their spinners until they accumulate $1000 or more. They should record how many spins it took them to do this. They should repeat this process three times. On the board, they can make a post-it note histogram. They will place three post-it notes each (one for each simulation). The x-axis should be the fewest number of spins to reach $1000 or more. They can then calculate the average of the simulations to discover the fewest number of spins (on average) it takes to accumulate $1000 or more. Students should discuss the following: the least number of spins, the most number of spins, the histogram’s center, shape, and spread. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 17 of 120 Mathematics I 1st Edition Unit 4 Notes on Spinner Learning Task 3 Give the spinner to the right to students. Ask them, “How is this spinner different from yesterday’s spinner?” They should reply, “The sections are not of the same area.” Next have them use their protractors to figure out the angle of each section and calculate the probabilities. 1) What is the probability of obtaining $800 on the first spin? $600 $400 $100 $300 $800 1/6 $700 $500 $200 2) What is the probability of obtaining $500 on the first spin? 1/12 3) Is it just as likely to land on $100 as it is on $800? No. The areas are not equal. 4) What is the probability of obtaining at least $500 on the first spin? 1/12 5) What is the probability of obtaining less than $200 on the first spin? 1/12 6) What is the probability of obtaining at most $500 on the first spin? 1/12 + 1/6 + 1/6 + 1/6 + 1/12 = 2/3 The following questions will be a little more difficult because there are two spins involved. You may want to draw a tree diagram and label the branches with the associated probabilities. Since each spin should be independent of each other, you can multiply the probabilities. 7) If you spin the spinner twice, what is the probability that you will have a sum of $200? (1/12)*(1/12) = 1/144 8) If you spin the spinner twice, what is the probability that you will have a sum of at most $400? You could have a (100, 100) or (100, 200) or (200, 100) or (200, 200). So, the answer is (1/12)(1/12)+2(1/12)(1/6)+(1/6)(1/6)=1/16 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 18 of 120 Mathematics I 1st Edition Unit 4 9) If you spin the spinner twice, what is the probability that you will have a sum of at least $1500? 1/18 10) Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200? 1/12 Comments: Students may be able to reason that the answer is 1/12 because the only way to get a sum of $200 is to spin ($100, $100). If you look at the smaller sample space where $100 was the first spin, then the question just becomes, “What is the probability of landing on $100?” If they do not get 1/12 by reasoning, then you can show them the following: Although these problems can be solved logically, you may want to introduce the formula for conditional probability if it helps some students. This formula will be used in the True or False Unit. P(A given B) = P(A and B)/P(B). It’s symbolically written as such P( A B) where stands for the intersection of sets A and B. P( A B) P( B) P(sum is $200 you land on $100 on the first spin) P(sum is $200 and the first spin is $100) P(the first spin is $100) P( sum is $200 and the first spin is $100) = 1/144 P(the first spin is $100) = 1/12 1 144 1 So, the answer is= 12 = 1/12 11) Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1500? (1/24)/(1/6)= ¼ 12) If you spin the spinner once, how much money, on average, would you expect to receive? (1/6)(800)+(1/12)(100)+(1/6)(400)+(1/12)(600)+1/6(300)+(1/12)(700)+(1/6)(200)+ (1/12)(500) = $441.67 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 19 of 120 Mathematics I Unit 4 1st Edition 13) If you spin the spinner twice, how much money, on average, would you expect to receive? $883.33 14) If you spin the spinner 10 times, how much money, on average, would you expect to receive? $4416.67 Comments: You may want to pursue one more investigative task if you have time. On average what is the fewest number of spins it will take to accumulate $1000 or more? To solve this, the students can simulate and use statistics to figure out the answer as seen below: Students spin their spinner until they accumulate $1000 or more. They should record how many spins it took them to do this. They should repeat this process 3 times. On the board, they will make a post-it note histogram. They will place 3 post-it notes a piece (one for each simulation.) The x-axis should be the fewest number of spins to reach $1000 or more. They will then calculate the average of the simulations to discover the fewest number of spins (on average) it takes to accumulate $1000 or more. A discussion about the least number of spins and the most number of spins should take place as well as the discussion of the histogram’s center, shape and spread. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 20 of 120 Mathematics I 1st Edition Unit 4 Notes on Spinner Learning Task 4 Give the spinner to the right to students. This spinner is different than yesterday’s spinner because the sections are not of the same area. You should have them use their protractors to figure out the angle of each section. It is also different because of the “bankrupt” section. This will make some probabilities more difficult to compute. Note: The angle measures are as follows: Bankrupt….50 degrees $800…..35 degrees $100…40 degrees $400…..50 degrees $500….45 degrees $200….55 degrees $300….40 degrees $600….45 degrees $400 $100 $800 $500 Bankrupt $200 1) What is the probability of obtaining $800 on the first spin? 35/360 = 7/72 2) What is the probability of obtaining $500 on the first spin? 1/8 3) Is it just as likely to land on $100 as it is on $800? No…$100 has a larger angle so it’s more likely 4) What is the probability of obtaining at least $500 on the first spin? (45+45+35)/360 = 25/72 5) What is the probability of obtaining less than $200 on the first spin? 1/9 6) What is the probability of obtaining at most $500 on the first spin? 23/36 (not including bankrupt) Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 21 of 120 $300 $600 Mathematics I Unit 4 1st Edition 7) If you spin the spinner twice, what is the probability that you will have a sum of $200? (1/9)*(1/9)=1/81 8) If you spin the spinner twice, what is the probability that you will have a sum of at most $200? 83/288 9) If you spin the spinner twice, what is the probability that you will have a sum of at least $1500? 49/5184 10) Given that you landed on $100 on the first spin, what is the probability that the sum of your two spins will be $200? 1/9 11) Given that you landed on $800 on the first spin, what is the probability that the sum of your two spins will be at least $1500? 7/72 12) If you spin the spinner once, how much money, on average, would you expect to receive? (5/36)(0)+(7/72)(800)+….=$362.50 Comment: Unlike the last two days, you may want to do some simulations to answer problems 13 and 14. If students land on bankrupt, they lose all of the money they have accumulated. 13) If you spin the spinner twice, how much money, on average, would you expect to receive? Explore through simulations 14) If you spin the spinner three times, how much money, on average, would you expect to receive? Explore through simulations Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 22 of 120 Mathematics I Unit 4 1st Edition 15) How many times on average would you have to spin until you land on “bankrupt?” Comments: Ask students to describe a way to solve this problem. Hopefully, they will describe an accurate way to use statistics and simulations. Have them perform the simulations and come up with the answers. Solution: Theoretically, the mean of a geometric distribution = 1/p. Therefore, the students should get about 1/(5/36) = 7.2. You should expect to have about seven spins until you land on bankrupt. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 23 of 120 Mathematics I 1st Edition Unit 4 Notes on Spinner Learning Task 5: A student created a spinner and records the outcomes of 100 spins in the table and bar graph below. Amount of Money on each $0 $100 $200 $300 $400 $500 $600 section of the spinner Number of times the 30 10 15 20 10 10 5 student lands on that section Student's outcomes from 100 spins 35 30 Frequency 25 20 15 10 5 0 Money per spinner section Based on the table and graph above, calculate the experimental probabilities of landing on each section of the spinner. Use these probabilities to draw what the spinner would look like below: $200 $100 The experimental probabilities are as follows: P($0) = 30/100 or .3, P($100)= 10/100 =.1, P($200) = .15, P($300)= .2, P($400) = .1, P($500) = .10, P($600)= .05 Calculate the average amount of money a person would expect to receive on each spin of the spinner. $300 $0 $400 0(.3) + 100(.1) +200(.15) + 300(.2) + 400(.1) + 500(.1) + 600(.05) = $220 Calculate the probability that you will receive at least $400 on your first spin. .25 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 24 of 120 $500 $600 Mathematics I 1st Edition Unit 4 Given you land on $300 the first time, what is the probability that the sum of your first two spins is at least $600? You can land on (300, 300), (300,400), (300, 500), or (300, 600). The answer is 4/7. What is the probability that the sum of two spins is $400 or less? For two spins, there are 7*7 or 49 possible outcomes. If the sum is $400 or less, then you can land on (0,0), (0, 100), (0, 200), (0, 300), (0, 400), (100, 100), (100, 200), (100, 300), (200, 200) (100, 0), (200, 0), (300, 0), (400, 0), (200, 100), or (300, 100). So, the probability is 15/49 You propose a game: A person pays $350 to play. If the person lands on $0, $200, $400, or $600, then they get that amount of money and the game is over. If the person lands on $100, $300, $500, then they get to spin again, and they will receive the amount of money for the sum of the two spins. In the long run, would the player expect to win or lose money at this game? If the player played this game 100 times, how much would he/she be expected to win or lose? Event P(Event) Money won 0 .30 $0 200 .15 $200 400 .10 $400 600 .05 $600 Event: P(event) Money won: (100, 0) (100,100) (100,200) (100,300) (100,400) (100, 500) (100, 600) (.10)(.30)=.03 (.1)(.1) = .01 (.1)(.15) = .015 (.1)(.2) = .02 (.1)(.1) = .01 (.1)(.1) = .01 (.1)(.05) = .005 $100 $200 $300 $400 $500 $600 $700 Event: (300, 0) (300, 100) (300, 200) (300, 300) (300, 400) (300, 500) (300, 600) P(Event) (.2)(.3) = .06 (.2)(.1) = .02 (.2)(.15) = .03 (.2)(.2) = .04 (.2)(.1) = .02 (.2)(.1) = .02 (.2)(.05) = .01 100* 300* 500* Expected amount money won if 100 is first:$32 Money Won: $300 Expected amount money if $400 300 is 1st: $104 $500 $600 $700 $800 $900 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 25 of 120 Mathematics I (500, 0) (500, 100) (500, 200) (500, 300) (500, 400) (500, 500) (500, 600) 1st Edition Unit 4 (.1)(.3)=.03 (.1)(.1)=.01 (.1)(.15) = .015 (.1)(.20)= .02 (.1)(.1) = .01 (.1)(.1) = .01 (.1)(.05) = .005 $500 $600 $700 $800 $900 $1000 $1100 Expected amount money if 500 is 1st: $72 Total Expected amount of money won: .3(0) + .15($200) + .1($400) +.05($600) + $32 + $104 + $72 = $308 Since the person payed $350 to play the game, then the person will lose about $42 each time he/she plays. If the player continues to play 100 times, then he/she is expected to lose 100($42) or $4200. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 26 of 120 Mathematics I Unit 4 1st Edition Notes on Spinner Learning Task 6 Design and make a workable spinner. Use your spinner to compare experimental vs. theoretical probabilities. Use your spinner to calculate expected values, probabilities of mutually exclusive events and conditional probabilities. Based on your comparisons of the experimental and theoretical probabilities, answer the question, “is your spinner fair?” The following rubric will be used to grade your spinner and paper. You must turn in the spinner along with a paper that describes the experimental and theoretical probabilities above. Comment: Make sure appropriate mathematics and mathematical vocabulary is used throughout the paper. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 27 of 120 Mathematics I 1st Edition Unit 4 Notes on Spinner Learning Task 7 Suppose you are playing “Wheel of Fortune” and you are the first player to spin the wheel. Is it likely that you will solve the puzzle (provided you never land on “bankrupt” or “lose a turn”)? In order to answer this question, calculate the following probabilities. Use those probabilities in your explanation. 1. 2. 3. Suppose you are the first player to spin the wheel, and you do not know the phrase, what is the probability that you guess a correct letter? If it turns out that no letters in the word are repeated, what is the probability that the 2nd letter you guess is correct (if you still do not know the phrase)? So, if you have a 5 letter word, and no letters in the word are repeated, what is the probability that you guess all 5 correct (provided that you still have no idea what the word is even after the 4th letter)? 1 which is very unlikely. 26 25 24 23 22 (However, it is also unlikely that a person would not know the word after the 4th letter was guessed correctly). The probability should equal P 26 25 24 23 22 and talk a little bit n! about permutations, the formula, n Pr , and how to use the calculator when (n r )! using this formula. Caution students that all probability problems do not require this formula. At this point, you may want to show that 4. 26 5 Suppose that you are the 3rd player to spin the wheel. You know that “S” and “T” are not in the phrase since those were the guesses of the first two players. What is the probability that you guess all 5 letters correctly if you never know what the word is? At this point, students will probably agree that it is difficult to win due to guessing alone. For the rest of the period, you should continue discussing permutations and combinations noting their similarities and differences. It is important for you to discuss the number of ways to rearrange letters if some letters repeat (such as in the word “Mississippi). This idea will lay the foundation for combinations. For homework, you may want to give classic counting problems involving permutations and combinations. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 28 of 120 Mathematics I Unit 4 1st Edition Notes on Spinner Learning Task 8 The students should bring to class a novel, newspaper, or magazine for this task. Comments: Today, your students will begin to come up with a strategy to increase their chances of winning in the bonus round of Wheel of Fortune using mathematics. You can watch Wheel of Fortune every weekday night on NBC at 7:30 pm to get ideas of how the bonus round is played. Play the bonus round with your students about 10 times. Or, divide your students into groups and let each group play 5 times (This would yield more data.). You may want to set a time limit of 3 minutes per phrase. This will be similar to playing, “hangman.” Suppose you are playing the bonus round on the game show, “Wheel of Fortune.” If you were allowed to pick any 8 consonants and any 2 vowels, which letters would you pick? In a small group, play the “bonus round” from “Wheel a Fortune.” This is a modified version of hangman. Let one member of your group come up with a phrase. You are to use only the 8 consonants and 2 vowels that you pick. Record the time it takes you to guess the phrase (do not take more than 3 minutes per phrase). Perform this simulation within your group 5 times. Record the length of time it took you to guess each of the 5 phrases. Collect the class data (time it took to guess the phrases). Plot the data on the board. Record the class data below. Show the students how to calculate the five number summaries: minimum, 1st quartile, median, 3rd quartile, and maximum from the 10 times. Also show them how to calculate the interquartile range (Q3-Q1). Then show them how to display their data (length of time to guess the phrase) in a boxplot. Calculate the 5 number summaries and draw a box plot. Are there outliers? What is the interquartile range? After you finish with the boxplot, record the times in a histogram and calculate the mean and the mean absolute deviation. Discuss with the students whether it is more appropriate to use the mean as the measure of center or the median as the measure of center. For example, if there were phrases that could not be guessed, then the time for those would be considered as outliers. If outliers are present, then the median tends to be a better measure of center. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 29 of 120 Mathematics I Unit 4 1st Edition Calculate the mean and mean deviation of the class data. Which is a better measure of center to use, the mean or the median? Why? Which is a better measure of spread to use, the interquartile range (IQR) or the mean deviation? Why? Discuss the variability of the times with the students. Have them describe what the IQR (interquartile range) represents in terms of percents (it’s the middle 50% of the data). Ask them what percent of the data is below the first quartile. Ask them what percent of the data is above the 3rd quartile. Now, use statistics to determine which letters and consonants are used the most in the English language. Prior to beginning this simulation, make a tally sheet. Write down all 26 letters to the alphabet in a vertical column on your notebook paper. Next, open a book/novel, close your eyes, and put your finger somewhere on the page. Begin at that spot and count how many A’s, B’s, C’s, etc. occur in the first 150 letters that they see. Tally on the notebook paper Comment: The students should open their novel, close their eyes, and put their finger somewhere on the page. This is where they will begin their simulation. Ask them about the importance of opening their books to a random page (especially if they all have the same book). Discuss why it is also important to find a random place to start on the page The students will count how many A’s, B’s, C’s, etc. occur in the first 150 letters that they see. They should place tally marks on their notebook paper to record this data. Based on your simulation, answer the following questions: Which 8 consonants and 2 vowels are used the most often in the English language? Compare your answer with your classmate. Did you pick the same letters? Write the class data below. Compute the percent of A’s, percent of B’s, percent of C’s, etc. from the class data. Compare your individual answer to the class answer. Answer the question, “which 8 consonants and 2 vowels are used most often in the English language?” Comment: The class should come to an agreement about the consonants and vowels. As they begin the next part of the task, students will be using the same consonants and vowels. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 30 of 120 Mathematics I Unit 4 1st Edition Using the 8 consonants and 2 vowels that are used most frequently in the English language, play the bonus round of wheel of fortune again within your group five times. Record the time it takes you to guess the phrase (do not take more than 3 minutes per phrase). The teacher will collect the class data. Using the class data, calculate the 5 number summary (minimum, 1st quartile, median, 3rd quartile, and maximum) and draw the associated box plot. Compare and contrast the two box plots (old box plot before we knew the most frequently used letters with this box plot). In your explanation, you should compare the centers (medians), the IQR’s (interquartile range), and the shapes (skewed or symmetric). You should then use these values to answer the following questions: Comments: The students should make dot plots of the two distributions and compute the mean and mean deviations of the times. Based on the dot plots and the box plots, they should determine whether the mean or median would be a better measure of center. They should also determine whether the IQR or mean deviation would be a better measure of spread/variability. (This decision mostly depends on outliers. Outliers affect the mean and mean deviation. They usually do not affect the median or IQR unless the sample size is small). 1. Did you save time today by using the letters we found to be used most often? Explain. 2. It took more than ______ seconds to answer 25% of the puzzles when we randomly provided the letters. It took more than _____ seconds to answer 25% of the puzzles when we used the most frequently used letters. 3. We answered 25% of the puzzles in less than _______seconds when we randomly guessed the letters. We answered 25% of the puzzles in less than _________ seconds when we used the most frequently used letters. 4. It took more than ________ seconds to answer half of the problems when we randomly provided the letters. It took more than ______ seconds to answer half of the problems when we used the most frequently used letters. 5. The bonus round only allows the player about 10 seconds to guess the phrase. Based on that, would we win more often or less often by randomly guessing or by using the frequently chosen letters? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 31 of 120 Mathematics I Unit 4 1st Edition Notes on Spinner Learning Task 9 Susan played the bonus round of wheel of fortune 30 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 17, 18, 19, 21, 24, 24, 24, 26, 28, 31, 33, 34, 35, 35, 37, 40 Monique also played to bonus round of wheel of fortune 25 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 55 a) Graph the two distributions below. Which measure of center (mean or median) is more appropriate to use and why? Calculate that measure of center. Susan’s Distribution: mean = 20.9333 median = 17.5 Monique’s Distribution: mean = 16.56 median = 15 Both distributions are skewed right; therefore, the median will be the better measure of center. Monique’s distribution has an outlier at 55. Without the outlier, the graph of the distribution will be approximately symmetric. Due to the outlier, the median is a better measure of center. b) Comment on any similarities and any differences in Susan’s and Monique’s times. Make sure that you comment on the variability of the two distributions. The IQR of Susan’s times is 28-13 = 15. The range of Susan’s times is 40-10 = 30. The IQR of Monique’s times is 16.5-14 = 2. The range of Monique’s times is 55-12 = 43. Although Monique had a larger range of scores than Susan, it was due to her outlier of 55. Most of the time, Monique was much more consistent than Susan. The middle 50% of her scores were within 2 seconds of each other. Whereas, the middle 50% of Susan’s scores were within 15 seconds of each other. c) If you are only allowed 15 seconds or less to guess the phrase correctly in order to win, which girl was more likely to win and why? 50% of Monique’s guesses were 15 seconds or less. Whereas, 50% of Susan’s guesses were 17.5 seconds or less. Therefore, Monique had a higher percentage of guesses less than or equal to 15 seconds. d) If Susan found that she could have guessed each phrase 3 seconds faster if she had chosen a different set of letters, would that have made any difference in your answer to part c? Why/why not? Yes, Susan’s median would become 17.5-3 or 14.5 seconds. Therefore, she would have at least 50% of her guesses less than 15 seconds. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 32 of 120 Mathematics I Unit 4 1st Edition Notes on Testing Learning Task 1 In the next series of tasks, students will calculate binomial probabilities. They will look at binomial distributions and calculate their means, medians, mean deviations, and interquartile ranges. They will also compute conditional probabilities and probabilities of dependent events. They should start to distinguish between dependent and independent events. They will graph probability distributions and calculate summary statistics from those distributions. Today you are going to determine how well you would do on a true/false test if you guessed at every answer. Take out a sheet of paper. Type randint(1,2) on your calculator. If you get a 1, write “true.” If you get a 2, write “false.” Do this 20 times. Comments: In the directions above, the students are using a calculator to randomly generate their ‘answers’ to the test. When they have finished, use your calculator to generate the answer key. Type “randint(1,2),” and call out the answers. Let “1” mean “true,” and let “2” mean “false.” Allow them to grade their “tests.” Before the teacher calls out the answers, how many do you expect to get correct? Why? Comments: Some should say 10. Ask them why? Since the probability of “true” is ½, they should say that they should get ½ of 20 correct. Grade your test. How many did you actually get correct? Did you do better or worse than you expected? Make a dot plot of the class distribution of the total number correct on your paper below. Calculate the mean and median of your distribution. Which measure of center should be used based on the shape of your dot plot? Comments: Discuss with the students which measure of center should be used based on the shape of the data. The mean should be a good measure because the dot plot is expected to be symmetrical (bell shaped) about 10. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 33 of 120 Mathematics I Unit 4 1st Edition Calculate the mean deviation and the IQR. What do these numbers represent? Which measure of variability should be used based on the shape of your dot plot? The mean deviation should be used when the mean is used as the center. The IQR should be used when the median is used as the center. The mean deviation is the average distance each point is from the mean. The IQR represents the range of the middle 50% of the points. Based on the class distribution, what percentage of students passed? Calculate the probabilities based on the dot plot: Solutions: Answers will depend upon class data. Students should use the dot plot to answer the following questions. 1. 2. 3. 4. 5. 6. 7. 8. What is the probability that a student got less than 5 correct? What is the probability that a student got exactly 10 correct? What is the probability that a student got between 9 and 11 correct (inclusive)? What is the probability that a student got 10 or more correct? What is the probability that a student got 15 or more correct? What is the probability that a student passed the test? Is it more likely to pass or fail a true/false test if you are randomly guessing? Is it unusual to pass a test if you are randomly guessing? Comments on the next set of questions: Review the binomial theorem that was taught earlier in the semester. It’s important to note the assumptions for a binomial distribution. They are as follows: Each trial has two outcomes….success “p” or failure “1-p.” There is a fixed number of trials “n.” The probability of success does not change from trial to trial. The trials are independent. The results for one trial does not depend on the results from another trial. Apply the binomial theorem to the binomial probability problems #1-5. Start with a few warm-up problems such as the ones below to explain how to calculate binomial probabilities. Additional Problem: Calculate the probability that the student got exactly 5 correct on the test. Explanation: If the student got 5 correct, then 15 were incorrect. P(correct) = ½ and the P(incorrect) = ½. So, if the first 5 were correct, and the last 15 were incorrect, then the student would have CCCCCIIIIIIIIIIIIIII graded on the quiz. The probability of Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 34 of 120 Mathematics I 1st Edition Unit 4 5 15 1 1 getting that in that order is . But, the problem did not specify that the first 5 2 2 were correct and the last 15 were incorrect. So that leads us to the question, “How many ways can we rearrange the 5 correct and 15 incorrect problems?” I usually remind students of how we rearranged the letters to the word “Mississippi.” Then, they 20! usually figure out that the answer is since there are five C’s that repeat and 5!15! fifteen I’s that repeat. So our final answer to the question, “What is the probability 5 15 20! 1 1 that the student got exactly 5 correct is . At this point, you can relate 5!15! 2 2 20! to 20 C5 on the calculator. Or, you can relate the values of n Cr to the values on 5!15! Pascal’s triangle. Additional Problem: Calculate the probability that the student got fewer than 3 correct on the test. Explanation: First you need to make sure that they understand that fewer than 3 means 0, 1, or 2 correct. Since these outcomes are mutually exclusive (if you get 0 correct, then you will not get 1 correct, etc.), then you can add the probabilities. So, you would calculate each of the following probabilities: 0 1 1 P(0 correct) = 20 C0 2 2 1 P(1 correct) = 20 1 or 2 20 19 1 1 20 C1 2 2 2 18 1 1 P(2 correct) = 20 C2 2 2 Once your students have mastered this reasoning and formula, then you may want to show them the calculator buttons “binompdf(n, p, x) and binomcdf(n, p, x).” On the TI-83 and TI-84, you can find “binompdf” under “2nd” , “VARS”, and then scroll down. If you want to calculate the probability of getting exactly 5 correct on the test, then you would type binompdf(20,1/2,5). If you want to calculate the probability of fewer than 3 are correct, then you would type binomcdf(20, ½, 2). Unlike the binompdf key, the binomcdf key adds up the probabilities starting at 0 and ending at x. The “c” stands for cumulative in binomcdf. Now the students should have enough information to calculate the theoretical probabilities of the following: Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 35 of 120 Mathematics I Unit 4 1st Edition Discuss: How do the theoretical probabilities compare to the experimental probabilities? 9. What is the probability that a student got less than 5 correct? Binomcdf(20,.5,4)=.0059 10. What is the probability that a student got exactly 10 correct? Binompdf(20, .5, 10)= .176 11. What is the probability that a student got between 9 and 11 correct (inclusive)? Binomcdf(20, .5, 11)- binomcdf(20, .5, 8) =.4966 12. What is the probability that a student got 10 or more correct? 1-binomcdf(20, .5, 9) =.588 13. What is the probability that a student got 15 or more correct? 1-binomcdf(20, .5, 14)=.0207 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 36 of 120 Mathematics I Unit 4 1st Edition Notes on Testing Learning Task 2: Comment: In the last task, we looked at true/false tests. In this task we will examine multiple choice tests. Suppose there is a 5 question multiple choice test. Each question has 4 answers (A, B, C, or D). If you are strictly guessing, calculate the following probabilities: (You may need to remind them how to do this from yesterday using the binomial theorem….note P(correct)=1/4 and P(incorrect)=3/4.) 0 1. P(0 correct) = 5 1 3 5 C0 = binompdf(5, .25,0)=.2373 4 4 2. P(1 correct) = .39551 3. P(2 correct) = .263367 4. P(3 correct) = .08789 5. P(4 correct) = .01465 6. P(5 correct) = .00098 Draw a histogram of the probability distribution for the number of correct answers. Label the xaxis as the number of correct answers. The y-axis should be the probability of x. The histogram should be skewed right Based on the distribution, how many problems do you expect to get correct? Comment: the mean of a binomial distribution is np or 5(.25)=1.25. From the graph, students should say that they expect one correct answer if guessing. Based on the distribution, how likely is it that you would pass if you were strictly guessing? (Calculate the probability of getting 4 or 5 correct.) P(4 or 5 correct) = .01465+.00098= .01563 So, it is not likely that you would pass by guessing alone. What is the probability that you will get less than 3 correct? .63281 What is the probability that you will get at least 3 correct? .10352 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 37 of 120 Mathematics I Unit 4 1st Edition Now let’s look at tests, such as the SAT, when you are penalized for guessing incorrectly. Suppose you have a multiple choice test with five answers (A, B, C, D, or E) per problem. Then, the probability your guess is correct = 1/5. And the probability that your guess is incorrect is 4/5. Suppose the test that you are taking will penalize you by ¼ of a point if you guess incorrectly. Test scores will be rounded. Suppose the test that you are taking will penalize you by ¼ of a point if you guess incorrectly. Scores will be rounded. If you strictly guess and get exactly 4 correct and 6 incorrect, what would be your score? 4-(.25)6 = 2.5, so your rounded score would be 3/10 or 30% If you take a 10 question test and know that 8 questions are correct, should you guess the answers for the other two questions? 8-.25(2) = 7.5 which rounds to 8. Yes, you have nothing to lose. If you take a 10 question test and know that 6 questions are correct, should you guess the answers for the other 4 questions? 6-.25(4) = 5. You could make a 50%; however, if you guessed one of the 4 correctly (which is expected), then you would still make a 60%. Given that you answered all 10 questions and you knew that 6 were correct, answer the following questions: If you can eliminate one of the answers for each of the 4 questions for which you are guessing, what would your percentage score be? The probability of guessing correctly for the 4 unknown problems is 1/4 since one answer is eliminated from 5 possible answers. Therefore, from the 4 questions which you did not know the answer, you expect 4(1/4)=1 to be correct and 4(3/4)=3 to be incorrect. So, your expected score should be 6 + 1 – (.25)(3) = 6.25 which rounds to 6 (or 60%). If you can eliminate two of the answers for each of the 4 questions for which you are guessing, what would your percentage score be? 6+ 4(1/3) – 4(2/3)(.25) = 6.666 which rounds to 7 (or 70 %) If you can eliminate three of the answers for each of the 4 questions for which you are guessing, what would your percentage score be? 6 + 4(1/2)-4(1/2)(.25)=7.5 which rounds to 8 (or 80%) Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 38 of 120 Mathematics I 1st Edition Unit 4 Notes on Testing Learning Task 3 Have you ever taken a multiple choice test when you may have had 4 or 5 “C’s” in a row and thought that you made a mistake? Did the teacher intentionally put 4 C’s in a row, or did you miscalculate? Or, could this randomly happen? This leads to the question we will answer today, “Can a person/teacher really be random with the questions when he makes up a 50 question True/False test?” Let’s try. On a piece of paper, you make up the answer key to a 50 question true/false test. All you need to do is randomly write down T (for true) or F (for false) such as: TTFFTF….etc. Now, on the bottom half of the paper, let your calculator generate your answer key. Type randint(0,1). “0” will stand for true, and “1” will stand for false. Press “enter” 50 times and record the outcomes of your calculator such as “00010110….”. Comment: Each student should have a string of 50 numbers. Of course, instead of “0” they may write “T” and instead of “1” they may write “F.” If you want to have some fun, you can select two or three volunteers from your class. Each student can give you his two lists. You can then use your “secret powers” to determine which one is the one he generated and which one is the one that the calculator generated. Chances are, the one that the calculator generated will have longer strings of “T’s” or “F’s” than the one that the student made up. After you’ve had your fun, then you can let them know that your “secret power” is really your statistical knowledge. Now you can let them in on the following secret by examining their data: Count the longest string of consecutive T’s that you recorded when you made up the answer key. For example, if you had FFTTTTTFFTTFFTF…., then your longest string of T’s may have been 5. Ask each student the length of their longest string of T’s. Make a dotplot of the distribution on the board. Find the center (mean or median) and the spread (mean deviation, IQR, or range). Now, count the longest string of consecutive T’s that you recorded when your calculator made up the answer key. Make a dot plot of the distribution on the board. Find the center (mean or median) and the spread (mean deviation, IQR, or range). Compare the two distributions. Does one distribution usually have a longer string of “T’s” than the other? On average, what is the longest string of “T’s” that you would expect to see on a true/false test if the answers were truly placed in random order? Comment: Theoretically, the expected longest run of “T’s” = log 1 (nq) where “p” is the p probability of being true, “n” is the number of questions, and “q” is the probability of Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 39 of 120 Mathematics I Unit 4 1st Edition being false. By plugging in the numbers, you should expect the longest string of “T’s” to be log 1 (50* 1 ) = 4.64. Practically, the list should have 4 or 5 T’s in a row at one 2 1 2 point in time. Now, do the same for a 50 question multiple choice test with 4 answers per problem (A, B, C, D). “How long would you expect the longest string of “C’s” to be?” Record guesses in a dot plot on the board. Comment: Many students think that teachers like the letter, “C” on multiple choice tests. Sometimes students will guess, “C” when they don’t know the answer. Use your calculator to randomly generate the answer key to the 50 question test. Enter, randint(1,4). Let “1” be “A”, “2” be “B”, “3” be “C”, and “4” be “D”. Each student should record their 50 answers and then count the longest string of “C’s” that they have. Make dot plot of this distribution on the board. Calculate and compare the center and spread of the two distributions. Is it likely that the teacher was random if he put 7 “C’s” in a row on a test? Is it likely that the teacher was random if he never put two consecutive letters in a row on a test? Comment: Using the formula above, you would expect the longest string of “C’s” to be 2.6144 (practically 2 or 3 in a row). Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 40 of 120 Mathematics I 1st Edition Unit 4 Notes on Testing Learning Task 4 A teacher makes up a 5 question multiple choice test. Each question has 5 answers listed “a-e.” A new student takes the test on his first day of class. He has no prior knowledge of the material being tested. a) What is the probability that he makes a 100 just by guessing? If the student makes a 100%, then he answered all 5 questions correctly. The P(correct) = 1/5. Therefore, the answer is (1/5)^5 = 1/3125 b) What is the probability that he only misses 2 questions? 3 2 1 4 5 C3 5 5 = .0512 If he misses 2 questions, then three are correct. P(3 correct) = c) What is the probability that he misses more than 2 questions? That means that he misses 3, 4, or 5 questions. Or, he gets 0, 1, or 2 correct. So, 0 5 1 4 2 3 1 4 1 4 1 4 C C C 5 0 5 1 5 2 5 5 5 5 5 5 = .94208 d) Let X= the number correct on the test. Make a graphical display of the probability distribution below. Comment on its shape, center, and spread. From a histogram, you should see that the shape is skewed to the right. The mean = np = 1 (which is one measure of center). The median is also 1. The IQR = 2-0 = 2. The range = 5. The mean deviation is .65536. e) Based on your distribution, how many questions should the new student get correct just by randomly guessing? One (since that occurs the most often and is also the mean). f) The answers to the test turned out to be the following: 1. A 2. A 3. A 4. A 5. C Do you think that the teacher randomly decided under which letter the answer should be placed when she made up the test? Explain. The student will probably use simulations to support his answer. If your students understood the log formula (which is not an objective at this level), then they could have shown that the longest string of letters in a row is about 1 for a 5 question test. Therefore, if the answers were randomly generated, you would not expect to have 4 A’s in a row. 4 log 1 5 log5 (4) .86135 5 1/ 5 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 41 of 120 Mathematics I Unit 4 1st Edition Notes on Testing Learning Task 5 Earlier, we found the probability that the student passed a multiple choice test just by random guessing. However, we know that students usually have a little more knowledge than that, even when they do not study, and consequently do not guess for all problems. Suppose that a student can retain about 30% of the information from class without doing any type of homework or studying. If the student is given a 15 question multiple choice test where each question has 4 answer choices (a, b, c, or d), then answer the following questions: 1. What is the probability that the student gives the correct answer on the test? What would be her percentage score on a 15 question test? 2. Given she provides the correct answer on the test, what is the probability that she strictly guessed? You may need to use the formula for conditional probability to do this problem. The formula is as follows: P(A given B) = P(A and B)/P(B). It’s symbolically written as such P( A B) where P( A B) P( B) stands for the intersection of sets A and B. Note: By solving the above formula, P( A B) P( A B) P( B) . With conditional probability problems, I have found tree diagrams very helpful. To solve the problem, I would make a tree diagram. The first branches would be P(knows answer) and P(does not know answer). The next branches would be P(correct given she knows answer), P(incorrect given she knows the answer), P(correct given she does not know the answer), and P(incorrect given she does not know the answer). The product of the two branches would be P(A and B)…P(knows answer and correct), P(knows answer and incorrect), P(doesn’t know answer and correct), P(doesn’t know answer and incorrect). The tree diagram should look like the one below. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 42 of 120 Mathematics I Unit 4 1 .3 0 1st Edition P(knows and correct)=(.3)(1)=.3 P(knows and incorrect)=(.3)(0)=0 .25 P(doesn’t know and correct)=(.7)(.25) .7 .75 P(doesn’t know and incorrect)=(.7)(.75) Using the tree diagram, and conditional probability formula, we can answer the following questions: What is the probability that the student gives the correct answer on the test? The student can give the correct answer when she knows it or when she doesn’t and guesses correctly. So the answer is (.3)(1) + (.7)(.25) = .475 What would be her percentage score on a 15 question test? 47.5% Given she provides the correct answer on the test, what is the probability that she strictly guessed? This is a conditional probability problem. You would use the following formula to solve P( guess and correct ) the problem: P( guess correct ) P(correct ) From the tree diagram P( guess and correct ) = (.7)(.25)= .175. Note: she would not guess if she knew the answer. We already calculated P(correct ) =.475. Therefore, P( guess correct ) = .175 .368. .475 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 43 of 120 Mathematics I 1st Edition Unit 4 Notes on Survey Learning Task Comment: Let the students work together to try to solve the following problem. A student conducted a survey with a randomly selected group of students. She asked freshmen, sophomores, juniors, and seniors to tell her whether or not they liked the school cafeteria food. The results were as follows: Liked food Did not like food Freshmen 85 44 Sophomores 50 92 Juniors 77 56 Seniors 82 78 Using the table above, calculate the following probabilities: 1. What is the probability that the randomly selected student was a freshman? 129/564 2. What is the probability that the randomly selected student was either a junior or senior? (133+160)/564 = 293/564 3. What is the probability that the randomly selected student was not a sophomore? 142/564 =422/564 4. If you knew that the student interviewed was a freshman, what is the probability that the student liked the cafeteria food? 85/129 5. If you knew that the student interviewed was a junior or senior, what is the probability that the student did not like the cafeteria food? (56+78)/(133+160)= 134/293 6. If you knew that the student did not like the cafeteria food, what is the probability that the student was not a freshman? 44/270 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 44 of 120 Mathematics I Unit 4 1st Edition Notes on Medical Learning Task Work in small groups to solve the following: A patient is tested for cancer. This type of cancer occurs in 5% of the population. The patient has undergone testing that is 90% accurate and the results came back positive. What is the probability that the patient actually has cancer? (For help, the question asks P(cancer given the test is positive)) P(cancer and tests positive) = .05(.90)= .045 P(test positive) = .045 + .95(.10) = .14 P(cancer given the test is positive) = .045/.14 = .3214 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 45 of 120 Mathematics I Unit 4 1st Edition Notes on Area Learning Task Let the students work together to try to solve the following problem: A circle is inscribed within a square having each side of length 2 units. A smaller square is inscribed within the circle such that the corners of the square intersect the circle. Note: The area of the large circle is 4. The area of the circle is (1)2 . The area of the small square is 2. Length of side = 2 1. If you throw a dart at the board, and it lands in the large square, what is the probability that it lands in the circle? Pi/4 = .78539 2. If you throw a dart at the board, and it lands in the large square, what is the probability that it lands in the small square. 2/4 3. If you throw a dart at the board and it lands in the circle, what is the probability that it does not land in the small square? (pi-2)/pi = .36338 Choose a point at random in the rectangle with boundaries 1 x 1 and 0 y 3 . This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x-coordinate and Y be the y-coordinate of the randomly chosen point. Find the following: a) P(Y>1 and X>0) = 2/6 b) P(Y>2 or X>0) = 1/6 c) P(Y>X) = 11/12 d) P(Y>2 given Y>X) =(2/6)/(11/12) = 4/11 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 46 of 120 Mathematics I Unit 4 1st Edition Notes on Card Learning Task Given a standard deck of 52 cards which consists of 4 queens, 3 cards are dealt, without replacement. 1. What is the probability that all three cards are queens? (4/52)*(3/51)*(2/50) = 24/132600 2. Let the first card be the queen of hearts and the second card be the queen of diamonds. Are the two cards independent? Explain. No, because if the 1st card is not replaced, then the second card drawn will have a different probability. 3. If the first card is a queen, what is the probability that the second card will not be a queen? 48/51 4. If the first two cards are queens, what is the probability that you will be dealt three queens? 2/50 5. If two of the three cards are queens, what is the probability that the other card is not a queen? 1 6. Answer questions #1 and #2 if each card is replaced in the deck (and the deck is well shuffled) after being dealt. #1. (4/52)(4/52)(4/52) = 64/140608 #2. Each draw should be independent of each other since the cards are replaced and well shuffled. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 47 of 120 Mathematics I Unit 4 1st Edition Notes on Marble Learning Task So far, we have looked at mostly independent events. For example, each spin of the spinner can be considered as independent because the outcome of the 2nd spin does not rely on the outcome of the first spin. The same can be said about rolling dice. Each roll is independent of each other. Today we will look at calculating dependent probabilities. When you sample without replacement, then the probabilities change. For example, suppose you have a deck of 52 cards. If I ask you what is the probability of drawing a queen, you would tell me 4/52. Now, suppose you drew a queen but did not replace the card. If I asked you “what is the probability of drawing a queen,” you would now tell me “3/51.” Note that this probability depends on the outcome of the first draw. Therefore, the two events are dependent. #1: There are 21 marbles in a bag. Seven are blue, seven are red, and seven are green. If a blue marble is drawn from the bag and not replaced, what is the probability that: a) A second marble drawn at random from the bag is blue? 6/20 b) A second marble drawn from the bag is blue or green? (6+7)/20 = 13/20 c) A second marble drawn from the bag is not blue? 1-6/20 = 14/20 #2: There are 21 marbles in a bag. Seven are blue, seven are red, and seven are green. If the marbles are not replaced once they are drawn, what is the probability: a) Of drawing a red marble and then a blue marble? (7/21)*(7/20)= 49/420 b) Of drawing a red marble, then a blue marble, then a green marble? (7/21)*(7/20)*(7/19) = 343/7980 c) Of drawing a red marble or a blue marble and then a green marble? (14/21)*(7/20) = 98/420 d) Of drawing a red marble given that the first marble drawn was red? 6/20 Comments: If events A and B are independent, the P( A B) P( A) P( B) . We can solve the conditional probability formula to read P( A B) P( A B) P( B) when events A and B are dependent or independent. If we want to test if A and B are independent , then we can use substitution to get, P( A) P( B) P( A B) P( B) . Hence, if A and B are independent, then P( A) P( A B) . Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 48 of 120 Mathematics I 1st Edition Unit 4 #3: Use the table below to answer the questions: A student conducted a survey with a randomly selected group of students. She asked freshmen, sophomores, juniors, and seniors to tell her whether or not they liked the school cafeteria food. The results were as follows: Liked food Did not like food Freshmen 85 44 Sophomores 50 92 Juniors 77 56 Seniors 82 78 a) What is the probability that a randomly selected student is a freshman? 129/564 b) What is the probability that a randomly selected student likes the food? 294/564 c) What is the probability that randomly selected student is a freshman and likes the food? 85/564 d) If the randomly selected student likes the food, what is the probability that he/she is a freshman? 85/294 e) Are the events “freshman” and “likes food” independent or dependent? They are dependent since P(freshman) = 129/564 and P(freshman given likes food) = 85/294 are not equal. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 49 of 120 Mathematics I Unit 4 1st Edition Notes on Dice Learning Task Comments: In the next few tasks, students will calculate probabilities based on rolling 2 or more dice. They will create and compare experimental to theoretical probability histograms. They will use the histograms and frequency tables to calculate probabilities. They will also learn how to perform simulations using their graphing calculators. Students should be familiar with most of these concepts from the Wheel of Fortune tasks. Like spinning a spinner, rolling dice are independent events. Therefore, calculating the probabilities will be similar. Students should learn how to transfer their knowledge from one context to another with this unit. Unlike the previous tasks, some of the sample spaces in the upcoming tasks are extremely large, especially when we use 5 dice. Therefore, the students will mostly rely on theoretical calculations to compute the probabilities. By the time we use 5 dice in this unit, we should have laid a good foundation with smaller sample spaces so the transition will not be too difficult. Materials Needed: Each student will need two dice…preferably of different colors. If dice are not available, then the graphing calculator can be used to simulate rolling two dice. 1. Roll the two dice (one red and one green) 100 times. Record your outcomes on a piece of notebook paper as below: Red Die Green Die Sum of Two Dice Tally how many times that you rolled a sum of 2, 3, 4, …, 12. Comments: Hopefully, by doing this, students will realize that (1, 2) is a different outcome than (2, 1) even though the sum for both is three. They should also realize that (2,2) is only one outcome. 2. Create a histogram based on the sums. The x-axis should be labeled “sum of two dice,” and the y-axis should be labeled “frequency.” 3. From your histogram, compute the mean and mean deviation. Comments: You may want to teach your students how to find the mean and the mean deviation on the TI-83 or TI-84 calculator to save some time in the future. A common mistake that students make when computing the mean and mean deviation of a frequency distribution is that they do not find the weighted average. They often Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 50 of 120 Mathematics I Unit 4 1st Edition average the numbers from 2 through 12 without multiplying them by their frequencies. Methods to do this on the calculator are described below: Method #1 for the mean: Enter the sums of the two dice (the numbers 2, 3, 4,…12) in List #1 (L1). Enter the frequency of the sums in L2. To find the mean, ask the students to highlight L3 and type in the formula L1*L2. (Note: To find L1 and L2, type 2nd and then 1 for L1. Type 2nd and then 2 for L2) Finally, go to the home screen and type sum(L3)/sum(L2). This should give you the mean of the frequency distribution. (To find the sum key, hit 2nd , stat, math, and then you should see sum listed under option #5.) Method #2 for the mean: This is much easier if the students really understand how to compute the mean of a frequency distribution by hand. Enter the data as described above into L1 and L2. Go to your home screen and select “stat”, “calc”, 1-Var Stats L1, L2. The mean should be the first value you see on the output. Method for mean deviation: You need to first calculate the mean as shown above. Highlight L3 to create the formula: abs(L1-mean). (Note: To find “abs,” enter “2nd” and then “0”. This gives you the catalog menu. “Abs” is the first option that you see.) On your home screen, type sum(L3)/sum(L2). This should be the mean deviation. 4. Pool class data. Graph the histogram for the class. What is the shape of the histogram? How does your histogram for your data compare to the class histogram of the class data? Compute the mean and the mean deviation of the class histogram and compare it to your summary statistics. Comments: On your overhead graphing calculator, type the numbers 2, 3, …, 12 in L1. You will type the class totals in L2. Graph the histogram for the students. Your histogram should look pretty symmetric. Ask students to sketch the class histogram on the same piece of paper as their histogram. Ask them to compare the two shapes (their histogram versus the class histogram). Compute the mean and the mean deviation of the class histogram. Ask the students to write these values on their paper next to the ones that they computed. Ask them to compare these numbers. Ask students to share their values and comparisons. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 51 of 120 Mathematics I Unit 4 1st Edition Discuss the importance of the size of the sample. The larger the sample size, the closer the experimental values get to the theoretical values. 5. Convert the y-axis of the class data from “frequency” to “probability.” Ask the students if there is any difference in the shape, center, or spread after the conversion is made. 6. Based on the class histogram (experimental probability), compute the following probabilities: 1) P( sum 5) = ________ 2) P( sum 4) =________ 3) P( sum 4) = __________ (You should remind students of using the complement) 4) P( sum 4 or sum 2) =_________ (remind students of mutually exclusive) 5) What is the probability that the sum = 4 if the first die was a 3?____ (remind students of conditional probability) 6) Now compute the theoretical probabilities of the sum of two dice. 7) Draw the theoretical probability distribution on your paper. The x-axis should be labeled “the sum of the two dice,” and the y-axis should be labeled as the probability (instead of the frequency). Find the mean, mean deviation, and the answers to the probability questions in #5 for the theoretical distribution. Compare the experimental and theoretical distributions. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 52 of 120 Mathematics I Unit 4 1st Edition Notes on Simulation Learning Task Instead of using real dice, today you will use the graphing calculator to simulate rolling 3 dice. On the TI-83 and TI-84, select the “Math” button. Next, select the “PRB” menu and choose “randInt”. On the home screen, randInt( , should appear. Type in the following: randInt(1,6,3). This will generate 3 random numbers between 1 and 6 inclusive. To roll again, just press “enter,” and three new random numbers between 1 and 6 will appear. Make a tally sheet for the sum of 3 dice. The minimum sum will be 3, and the maximum sum will be 18. Use your calculator to simulate rolling 3 dice 100 times. Record the sums on your tally sheet. Make a frequency distribution and find the mean, mean deviation, median, and IQR. Pool the class data on the board or the overhead calculator and make a class frequency distribution. Calculate the class mean, mean deviation, median and IQR. Discuss what these numbers represent. Use the class distribution to answer the following probability questions: Solutions will depend upon class data. 1. 2. 3. 4. 5. 6. P( sum 5) = ________ P( sum 4) =________ P( sum 4) = __________ P( sum 4 or sum 3) =_________ What is the probability that the sum = 6 if the first die was a 3?____ What is the probability that the sum = 12 if the sum of the first two dice is 10?___ Calculate the theoretical probabilities of obtaining a sum of 3, 4, 5, …., 18. On the same piece of paper, construct a theoretical probability distribution. Calculate the mean, mean deviation, median and IQR and compare it to the experimental probability distribution. Using the theoretical probabilities, compute the answers to the same questions above. Compare the answers. 7. P( sum 5) = 6/216 = 1/36 8. P( sum 4) = 4/216 = 1/54 9. P( sum 4) = 212/216 = 53/54 This would be a good time to discuss using complements. 10. P( sum 4 or sum 3) = 213/216 Remind them of mutually exclusive. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 53 of 120 Mathematics I Unit 4 1st Edition 11. What is the probability that the sum = 6 if the first die was a 3? 1/18 Remind them of conditional probability. 12. What is the probability that the sum = 12 if the sum of the first two dice is 10? 1/6 Now compute the following probabilities (if you roll 3 dice): 13. What is the probability of getting three 1’s on the first roll? 1/216 14. What is the probability of getting three of a kind on the first roll? 6/216 = 1/36 15. What is the probability of getting two 1’s and another number on the first roll (in any order)? 15/216 = 5/72 16. What is the probability of getting two of a kind (3rd dice must be different)? 6(5/72) = 30/72 17. What is the probability of getting three consecutive numbers (but in any order) on the first roll? 24/216 = 1/9 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 54 of 120 Mathematics I 1st Edition Unit 4 Notes on Game 1 Learning Task #1 Ask students if the following game is fair. Let students work together before you show them how to determine if it’s fair or not. A person pays $2 to play a game. He rolls two dice for this game. If he rolls an even sum, he wins $2.50 and goes home. If he rolls a sum of 3, 5, or 7, then he loses and goes home. If he rolls a sum of 9 or 11, he rolls again. If on his second roll, he rolls a sum of 9 or 11, he wins $5.00; otherwise, he loses and goes home. Amount Won $2.50 $0 $5.00 P(event) ½ 1/3 1/36 $0 5/36 Event Even sum Sum of 3, 5, or 7 Sum of 9 or 11. Roll again and win Sum of 9 or 11. Roll again and lose Expected Winnings = 2.50(1/2) + 0(1/3) + (5.00)(1/36)+ 0(5/36)= $1.39 Net Winnings= $1.39-$2.00 = -$0.61. #2: Create your own game in groups using 3 dice. Calculate the amount a person is expected to win or lose each time he plays the game. The students need to make the game so that its not much more likely to win as to lose. After the students have had an opportunity to make up the game and compute the expected values, they should play the game several times. Who won more times? Was that what was expected? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 55 of 120 Mathematics I Unit 4 1st Edition Notes on Game 2 Learning Task Pass out the yahtzee board game for students to play. Roll 5 dice. On the first roll, record whether or not you get the following: Only 3 of a kind: Only 4 of a kind: Full House (3 of a kind and 2 of a kind): Small Straight (sequence of 4 in any order): Large Straight (sequence of 5 in any order): Yahtzee (5 of a kind): None of the above: Ask them to record their data. Is it likely to get any of the above on the first roll? Which is most likely? How many points are awarded to this outcome? Why? Which is the least likely outcome? How many points are awarded to this outcome? Why? Now simulate rolling 5 dice on your calculator by entering randInt(1,6,5). Roll 5 dice 100 times each. Make a tally sheet to record the following: Only 3 of a kind: Only 4 of a kind: Full House (3 of a kind and 2 of a kind): Small Straight (sequence of 4 in any order): Large Straight (sequence of 5 in any order): Yahtzee (5 of a kind): None of the above: Pool the class data to calculate the experimental probabilities of the above outcomes. Work in small groups to calculate the theoretical probabilities. Comment: You may need to help them calculate at least one of the above theoretical probabilities. For example, what is the probability of rolling only 3 of a kind? Comments: That means we have 3 of the same number but the other two numbers must be different or it would we would have a full house or 4 of a kind. If the 1st three dice landed on the same number and the last two dice were different, then we would have 6*1*1*5*4 possibilities (6 choices for the 1st die but the second and third die would have to be the same as the first…so only 1 choice for those…then there are 5 numbers left to choose for the fourth die and then 4 choices left for the fifth die). However, the dice do Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 56 of 120 Mathematics I Unit 4 1st Edition not have to land in any specified order to get “three of a kind,” so that leads us to the question, 5! “how may ways can we rearrange 5 dice if 3 of them are the same?” The answer is (Think 3! of the number of ways to rearrange the letters “aaabc”.) So, the total number of possible ways 5! to get 3 of a kind is (6*1*1*5* 4) . Therefore, the probability of getting 3 of a kind is 3! 5! (6*1*1*5* 4) 3! =. 65 5! (6 1 1 1 5) 150 4! Only 4 of a kind: =25/1296 5 6 7776 5! (6 11 5 11) 300 2!3! Full House (3 of a kind and 2 of a kind): =25/648 5 6 7776 Only a Small Straight (sequence of 4 in any order): Note: This cannot be a large straight. Therefore, for the 5th die, one of the numbers in the sequence must repeat. Or, it must be the other number that’s not in the straight (example of an outcome: 1, 2, 3, 4, 5, 6). 5! (3 111 4) 5!(3 111 1) 720 2! = 5/54 5 6 65 7776 5!(2 1 1 1 1) 240 Large Straight (sequence of 5 in any order): = 5/162 65 7776 Yahtzee (5 of a kind): 6/7776 =1/1296 None of the above: 1-sum of the above = 55/108 #2: Students should calculate the expected score on the bottom half of their yahtzee score sheet. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 57 of 120 Mathematics I Unit 4 1st Edition Notes on Game 3 Learning Task Two players play a game. The first player rolls a pair of dice. If the sum is 6 or less, then player 1 wins. If it’s more than six, then player 2 gets to roll. If player 2 gets a sum of 6 or less, then he loses. If player 2 gets a sum greater than 6, then he wins. a) Which player, player 1 or player 2, is more likely to win? Why? Player 1 is more likely to win. The P(player 1) wins is 15/36. The P(player 2) wins is (21/36)(21/36) = 49/144. Player 2 only wins if player one does not win on the first roll and player two rolls a sum greater than 6. Player 2 could either lose on Player 1’s first roll, or player 2 could get a chance to roll and then roll a sum of 6 or less and lose. The probability that player 2 gets to roll and loses is (21/36)(15/36)=35/144. b) If player 1 is awarded 10 tokens each time he/she wins the game, how many tokens must player 2 be awarded in order for this to be a fair game? Why? You must solve the following equation in order to get the answer: 15 49 (10) ( x) So, player 2 must be awarded 12.244 tokens to be fair. If he’s awarded 36 144 12 tokens, the game is slightly in player 1’s favor. If he’s awarded 13 tokens, the game is slightly in player 2’s favor. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 58 of 120 Mathematics I Unit 4 1st Edition Notes on Culminating Task A student rolled 3 dice 100 times, found the sums of the 3 dice, and put them into the following frequency distribution: Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Frequency 1 3 5 4 8 9 13 15 12 13 7 3 6 1 2 0 Experimental Probability .01 .03 .05 .04 .08 .09 .13 .15 .12 .13 .07 .03 .06 .01 .02 0 a) Based on the student’s simulation, compute the experimental probabilities for the sum of 3 dice and write them in the table above. See the table above. b) Based on the student’s simulation, what is the expected value (the mean) of the sum of the three dice? 3(.01) + 4(.03) + …..18(0) = 10 c) Based on the student’s simulation, what is the median sum of the three dice? The median is also 10. d) Comment on the relationship between the mean and median relative to the shape of the distribution. The distribution is approximately symmetric. Therefore, the mean should be close to the median. In this case, they are the same. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 59 of 120 Mathematics I Unit 4 1st Edition e) Based on the student’s simulation, what is the probability that the sum of 3 dice is even? .03 + .04 + …. + 0 = .48 Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Theoretical probability P(x) 1/216 3/216 6/216 10/216 15/216 21/216 25/216 27/216 27/216 25/216 21/216 15/216 10/216 6/216 3/216 1/216 f) Based on the theoretical probabilities in the table to the left, what is the expected value (the mean) of the sum of the three dice? 3(1/216) + 4(3/216) + …. + 18(1/216) = 10.5 g) Based on the theoretical probabilities, what is the median sum of the three dice? The median is also 10.5. h) Comment on the relationship between the mean and median relative to the shape of the distribution. The theoretical distribution is perfectly symmetric (bell- shaped). Therefore, the mean = median. i) Based on the theoretical probabilities, what is the probability that the sum of 3 dice is even? 3/216 + 10/216 + … + 1/216 = .5 j) How does the theoretical probability that the sum of 3 dice is even compare to the experimental probability that the sum of 3 dice is even (part e). They are pretty close. The experimental probability was .48 for only 100 rolls. The theoretical is .5. k) How does the theoretical mean and median compare to the experimental mean and median from the student’s simulation? Again, they are pretty close. The theoretical mean and median is 10.5; whereas, the experimental mean and median is 10. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 60 of 120 Mathematics I Unit 4 1st Edition l) Display the experimental probability distribution and the theoretical probability distribution graphically so that they can be easily compared. The student should draw side to side boxplots or histograms. If you look at the boxplots, the experimental distribution has a smaller IQR than the theoretical distribution. The median is also slightly smaller for the experimental distribution. m) Based on your answers to parts “j, k, and l” above, do you think that the student really simulated rolling 3 dice 100 times, or did the student make up the data. Explain. The student’s explanation should include a direct comparison to the experimental vs. theoretical summary statistics. He/she should also use the concept of “randomness” in the explanation. An argument for a student who believes that the student did not fabricate the data might include: It would be unrealistic to get a perfectly symmetric distribution from the experimental data for only 100 rolls. But, a person should expect a somewhat symmetric distribution from rolling 3 dice. If it was perfectly symmetric or strongly skewed, it would be unlikely that the distribution was generated from the random event of rolling 3 dice. An argument for a student who believes that the data was fabricated might include: Since the mean is exactly the same as the median for the experimental data, then it is unlikely to be generated by a rolling 3 dice. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 61 of 120 Mathematics I Unit 4 1st Edition Additional Tasks that can be used as Culminating Tasks Culminating Task #2 Wheel of Fortune…Let’s Play to Win Susan played the bonus round of wheel of fortune 30 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 17, 18, 19, 21, 24, 24, 24, 26, 28, 31, 33, 34, 35, 35, 37, 40 Monique also played the bonus round of wheel of fortune 25 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 55 a) Graph the two distributions below. Which measure of center (mean or median) is more appropriate to use and why? Calculate that measure of center. b) Comment on any similarities and any differences in Susan’s and Monique’s times. Make sure that you comment on the variability of the two distributions. c) If you are only allowed 15 seconds or less to guess the phrase correctly in order to win, which girl was more likely to win and why? d) If Susan found that she could have guessed each phrase 3 seconds faster if she had chosen a different set of letters, would that have made any difference in your answer to part c? Why/why not? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 62 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #1 True/False Unit A teacher makes up a 5 question multiple choice test. Each question has 5 answers listed “a-e.” A new student takes the test on his first day of class. He has no prior knowledge of the material being tested. a) What is the probability that he makes a 100 just by guessing? b) What is the probability that he only misses 2 questions? c) What is the probability that he misses more than 2 questions? d) Let X= the number correct on the test. Make a graphical display of the probability distribution below. Comment on its shape, center, and spread. e) Based on your distribution, how many questions should the new student get correct just by randomly guessing? f) The answers to the test turned out to be the following: 2. A 3. A 4. A 5. A 6. C Do you think that the teacher randomly decided under which letter the answer should be placed when she made up the test? Explain. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 63 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #2 True or False Unit Given a standard deck of 52 cards which consists of 4 queens, 3 cards are dealt, without replacement. 1. What is the probability that all three cards are queens? 2. Let the first card be the queen of hearts and the second card be the queen of diamonds. Are the two cards independent? Explain. 3. If the first card is a queen, what is the probability that the second card will not be a queen? 4. If the first two cards are queens, what is the probability that you will be dealt three queens? 5. If two of the three cards are queens, what is the probability that the other card is not a queen? 6. Answer questions #1 and #2 if each card is replaced in the deck (and the deck is well shuffled) after being dealt. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 64 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #1 Yahtzee Unit Two players play a game. The first player rolls a pair of dice. If the sum is 6 or less, then player 1 wins. If it’s more than six, then player 2 gets to roll. If player 2 gets a sum of 6 or less, then he loses. If player 2 gets a sum greater than 6, then he wins. a) Which player, player 1 or player 2, is more likely to win? Why? b) If player 1 is awarded 10 tokens each time he/she wins the game, how many tokens must player 2 be awarded in order for this to be a fair game? Why? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 65 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #2 Yahtzee Unit A student rolled 3 dice 100 times, found the sums of the 3 dice, and put them into the following frequency distribution: Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Frequency Experimental Probability 1 3 5 4 8 9 13 15 12 13 7 3 6 1 2 0 a) Based on the student’s simulation, compute the experimental probabilities for the sum of 3 dice and write them in the table above. b) Based on the student’s simulation, what is the expected value (the mean) of the sum of the three dice? c) Based on the student’s simulation, what is the median sum of the three dice? d) Comment on the relationship between the mean and median relative to the shape of the distribution. e) Based on the student’s simulation, what is the probability that the sum of 3 dice is even? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 66 of 120 Mathematics I Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Unit 4 Theoretical probability P(x) 1/216 3/216 6/216 10/216 15/216 21/216 25/216 27/216 27/216 25/216 21/216 15/216 10/216 6/216 3/216 1/216 f) 1st Edition Based on the theoretical probabilities in the table to the left, what is the expected value (the mean) of the sum of the three dice? g) Based on the theoretical probabilities, what is the median sum of the three dice? h) Comment on the relationship between the mean and median relative to the shape of the distribution. i) Based on the theoretical probabilities, what is the probability that the sum of 3 dice is even? j) How does the theoretical probability that the sum of 3 dice is even compare to the experimental probability that the sum of 3 dice is even (part e). k) How does the theoretical mean and median compare to the experimental mean and median from the student’s simulation? l) Display the experimental probability distribution and the theoretical probability distribution graphically so that they can be easily compared. m) Based on your answers to parts “j, k, and l” above, do you think that the student really simulated rolling 3 dice 100 times, or did the student make up the data. Explain. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 67 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit Name:________________________ Sample Homework Day 1 1. The large circle has radius of 7 units. Each of the four smaller circles have radius of 1 unit. A point is chosen at random from inside the circle. What is the probability that the point will not be in one of the four smaller circles? 2. A square is formed by the intersection of the lines x=2, x=-1, y=-1, and y=2. A point is randomly chosen from inside the square. a) What is the probability that the point lies above the line y=x? b) What is the probability that the point lies above the line y 2 2 x ? 3 3 3. Given the spinner: 100 400 200 300 a) b) c) d) e) What is the probability of landing on 400? What is the probability of landing on a number less than 300? What is the probability of landing on a number greater than 100? If the spinner is spun twice, what is the probability that the sum is 500? If the spinner is spun twice, what is the probability that the sum is less than 500? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 68 of 120 Mathematics I Wheel of Fortune Unit Sample Homework Day 2 1st Edition Unit 4 Name:________________________ Assume all events described in the problems below are independent. 1. Sam makes 70% of his free throws. What is the probability that a) Sam makes 2 free throws in a row? b) Sam makes his first free throw and then misses his next free throw? c) Sam misses his first free throw and then makes his next? d) Sam misses both free throws? e) Sam makes at least one free throw? 2. If you roll a die, the probability that you land on a “one” is 1/6. If you roll the same die twice, what is the probability that: a) You land on two “ones” in a row? b) You never land on a “one”? c) You land on at least one “one”? 3. A sack contains 7 blue, 3 red, and 2 green marbles. If you draw a marble, replace it, and then draw another marble, what is the probability that: a) You draw a red marble and then a green marble? b) You draw a red marble and then another red marble? c) You draw at least one blue marble? d) You draw at least one red or blue marble? e) You never draw a blue marble? f) Which is more likely to occur and why? To draw the marble colors RBBR or to draw the marble colors BBGRBB? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 69 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit Sample Homework Day 3 Name:________________________ 1. Given the probability distribution, calculate the expected value (the mean). x P(x) 0 .3 1 .2 2 .4 3 .1 2. Suppose the probability of winning a game is 1/100. In order to play the game, you must pay $5. If you win, you will receive $200. Outcome P(outcome) Payoff Win 1/100 $200 Lose a) Fill in the probability distribution above. b) How much money do you expect to win each time you play the game (do not take into account the $5.00). c) What is your “net” loss each time that you play? d) If you played the game 50 times, how much money would you win/lose? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 70 of 120 Mathematics I Unit 4 Wheel of Fortune Unit Sample Homework Day 5 1st Edition Name:__________________ Note: The homework for day 4 should be to work on the project. 1. How many ways are there to rearrange all of the letters a) To the word “cat?” b) To the word “cat” if “c” must be the first letter of each arrangement? 2. How many ways are there to rearrange all of the letters a) To the word “book”? b) To the word “book” if “k” must be the first letter of each arrangement? c) To the word “book” if “o” must be the first letter of each arrangement? 3. How many ways to rearrange all of the letters a) To the word “Mississippi?” b) To the word “Mississippi” if “M” must be the first letter and “P” must be the last letter of each arrangement? 4. If each of four students shakes hands with each of the other 3 students one time, how many handshakes were made? 5. A locker has a combination lock with 40 numbers on it. Three numbers must be selected in the exact order or the lock will not open. How many combinations are possible if numbers can be repeated? How many combinations are possible if numbers cannot be repeated? 6. Six people are seated at a circular table for dinner. The hostess tries to determine the best way to seat everyone. How many different ways can she seat these six people? If there are 3 men and 3 women, How many different ways can she seat these six people if she wants to have men sitting next to only women? 7. An addition was made to an office building which provided 6 new offices. Currently, there are 10 staff members of equal ability who need offices. a) How many ways can the office manager assign the 10 staff members to the 6 new offices (obviously, 4 staff members will still not have offices)? b) There are 5 male staff members and 5 female staff members. The office manager decides that 3 males and 3 females must be placed in the 6 offices. How many ways can this be done? 8. A true/false test has 10 questions. Five answers are “true” and five answers are “false.” How many ways can you rearrange the answers “TTTTTFFFFF”? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 71 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit Sample Homework Day 6 Name:________________________ 1. A class had the following test averages on their 8 unit tests: 93, 82, 91, 64, 78, 90, 86, 87 a) b) c) d) e) f) g) h) i) j) k) Write the 5 number summary below: Compute the interquartile (IQR) range: What percent of data is contained in the IQR? Draw the boxplot. Describe the shape of the boxplot. Compute the mean of the class test averages. Compute the mean deviation of the class averages. Which measure of center is more appropriate to use and why? 25% of scores are at or below ________. 25% of scores are at or above ________. 50% of scores are between ______ and ______ or ______ and _______ or _____ and _____. 2. A student had 5 homework grades: 68, 99, 72, 100, and another one that he lost. The teacher told him that his homework average is 82. What was the 5th homework grade? 3. A student has taken 4 tests. Her test average is 82. What must the score be on her next test if she wants to raise her average to 83? 4. Given the numbers the consecutive integers 1, 2, 3,…., 31. What is the mean? What is the median? 5. If the median of a list of 31 consecutive integers is 42, what are the least and greatest integers in the list? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 72 of 120 Mathematics I Unit 4 Wheel of Fortune Unit Sample Homework Day 7 1st Edition Name:__________________ Sue and John had the following test scores: Sue’s scores: 79, 77, 78, 80, 80, 81, 83, 82, 80 John’s scores: 65, 60, 68, 80, 81, 92, 100, 95, 79 1. Calculate Sue’s five number summary: 2. Calculate John’s five number summary: 3. Draw Sue’s and John’s boxplots next to each other. 4. Compare their distributions. 5. Calculate Sue’s mean and mean deviation: 6. Calculate John’s mean and mean deviation: 7. Compare Sue’s and John’s averages and variability. 8. Which measure of center is more appropriate to use? Why? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 73 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit Sample Homework Day 8 Name:________________________ 1. Janet and Ed had the following scores: Janet’s scores: 50, 90, 70, 75, 80, 80, 90, 85, 90, 90 Ed’s scores: 75, 75, 80, 90, 95, 80, 80, 75, 75, 75 Compare Janet’s and Ed’s scores. Make sure that you display the data graphically. Calculate the summary statistics and use them when you discuss the similarities and differences in the distributions’ centers, shapes, and variabilities. 2. If 3 points were added to each of Janet’s scores, how would the following statistics change? a) b) c) d) e) f) g) Mean:_________________________ Median:__________________________ Range:_________________________ IQR:__________________________ Mean Deviation:__________________ Q1 (lower quartile):__________________ Q3 (upper quartile):________________________ 3. If each of Ed’s scores were multiplied by 1.1, how would the following statistics change? a) Mean:_________________________ b) Median:__________________________ c) Range:_________________________ d) IQR:__________________________ e) Mean Deviation:__________________ f) Q1 (lower quartile):__________________ g) Q3 (upper quartile):________________________ 4. Give a list of 5 numbers that are not equal to zero such that the mean deviation of those 5 numbers is equal to zero. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 74 of 120 Mathematics I 1st Edition Unit 4 True or False Sample Homework Day 1 Name:________________________ 1. How many ways are there to rearrange the letters ABBBAAAB? 2. How many different ways are there for a coin that is flipped 8 times to land on 3 tails and 5 heads? 3. How many different ways are there to choose a committee of 3 students from 10 students? 4. How many different ways are there to choose a president, vice-president, and secretary from 10 students (if a student cannot hold more than one office)? 5. What is the difference between question #3 and #4? 6. If you roll 3 dice, how many ways can exactly one of the dice land on 6? 7. If you roll 3 dice, how many ways can exactly two of the dice land on 6? 8. Write a question that is solved by the formula 5 P2 . 9. Write a question that is solved by the formula 5 C2 . 10. How many different hands of 5 cards are there in a deck of 52 cards? 11. A bag contains 6 yellow marbles, 3 blue marbles and 8 red marbles. How many ways are there to choose 3 yellow, 1 blue, and 2 red marbles? 12. If you randomly select 6 marbles from the bag, what is the probability that you select 3 yellow, 1 blue, and 2 red marbles (if the bag contains 6 yellow, 3 blue, and 8 red marbles)? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 75 of 120 Mathematics I 1st Edition Unit 4 True or False Unit Sample Homework Day 2 Name:__________________ 1. Suppose you guess on a multiple choice test where each question has 4 answers. a) What is the probability of guessing correctly? b) If there are 5 questions, what is the probability that you get all questions correct by guessing? c) If there are 5 questions, what is the probability that you get 4 or less correct by guessing? d) If there are 5 questions, what is the probability that you get 2 incorrect? e) If there are 5 questions, what is the probability that you get at most 2 incorrect? 2. Matthew is a 70% free throw shooter. Let x = # shots that Matthew makes. Let p(x) represent the probability of making that many shots. Matthew shoots 6 free throws. a) Fill in the probability distribution below: X 0 1 2 3 4 5 6 P(x) b) Draw a histogram of the probability distribution below: c) Describe the shape, center and spread of the distribution: d) Using your distribution in (a) or (b), what is the probability that Matthew makes at least 4 free throws? e) What is the probability that he misses at least 4 free throws? f) How many free throws do you expect Matthew to make? 3. Suppose a rocket is designed to have 50 important parts. If any one of these parts fail, then the rocket will crash. Therefore, each part was designed to be 99% reliable. Assuming that each part operates independently, then what is the probability that a) All of the parts work and the rocket does not crash? b) Only one of the parts does not work? c) How reliable must the parts be in order to insure that there is a 95% probability of all 50 parts working? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 76 of 120 Mathematics I True or False Unit Sample Homework Day 3 X P(x) 1st Edition Unit 4 Name:__________________ 1. Suppose a child born to a certain pair of parents has probability of .75 of having blood type O. These parents have 6 children. a) Let X= the number of children who inherit blood type O. Fill in the probability distribution below: 0 1 2 3 4 5 6 b) Draw a histogram of the probability distribution below: c) Describe the shape, center and spread of the distribution: d) Using your distribution in (a) or (b), what is the probability that at least 3 children will inherit blood type O? e) What is the probability that fewer than two children will not inherit blood type O? f) How many children do you expect to inherit blood type O? 2. Suppose a blood test given to a woman when she is 12 weeks pregnant is 90% accurate in detecting Downs Syndrome. a) Suppose 10 women take the test when they are 12 weeks pregnant. What is the probability that all 10 tests give an accurate reading? b) What is the probability that at least 9 of the 10 tests give an accurate reading? c) What is the probability that at least 7 of the 10 tests are accurate? d) Let X= the number of accurate tests. Draw a histogram of the probability distribution below. e) Describe the center, shape, and spread of the distribution. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 77 of 120 Mathematics I Unit 4 True or False Unit Sample Homework Day 5 1st Edition Name:__________________ 1. You survey a random group of students to find out which subject they like the most. You organized their responses into the table below: Math Science Language Arts Social Studies Freshmen 21 18 32 29 Sophomores 33 35 21 19 Juniors 25 28 22 32 Seniors 28 25 21 19 a) b) c) d) e) f) g) h) i) How many students participated in the survey? What is the probability that a randomly selected student liked math the most? What is the probability that a randomly selected student liked science the most? What is the probability that a randomly selected student liked Language Arts the most? What is the probability that a randomly selected student liked Social Studies the most? Of the freshmen, what is the probability that their favorite subject was either math or science? Of the sophomores, what is the probability that their favorite subject was neither math or science? Of the juniors, what is the probability that their favorite subject was either math or social studies? Of the seniors, what is the probability that their favorite subject was neither math or social studies? 2. A student knows about 25% of the material. The student takes a 10 question multiple choice test. Each question has 4 answers. a) What percent of the questions do you expect the student to answer correctly? b) If the student answers correctly, what is the probability that she guessed? 3. A company car was involved in a hit and run accident at night. 70% of the company cars are green and 30% of the company cars are blue. A witness at the scene identified the car in the accident as blue. The witness was tested under similar visibility conditions and correctly identified colors 80% of the time. What is the probability that the company car involved in the accident was blue? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 78 of 120 Mathematics I Unit 4 True or False Unit Sample Homework Day 6 1st Edition Name:__________________ 1. Choose a point at random in the rectangle with boundaries 1 x 1 and 0 y 3 . This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x-coordinate and Y be the y-coordinate of that point chosen. Find the following, and justify all answers numerically and graphically (or both): a) b) c) d) P(Y 1 and X 0) P(Y 2 or X 0) P (Y X ) P(Y 2 Y X ) 2. A point is randomly selected in the large 8 x 8 square below. Within the 8 x 8 square, there is a circle with radius 2. a) Find the probability that the point selected is inside the circle. b) Find the probability that the point selected is inside the smaller square. c) Given the point is inside the circle, what is the probability that it is inside the smaller square? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 79 of 120 Mathematics I 1st Edition Unit 4 True or False Unit Sample Homework Day 7 Name:__________________ 1. A bag contains 3 red marbles, 5 green marbles, and 2 blue marbles. Two consecutive draws are made from the bag without replacement of the first draw. Find the probability of each event: a) b) c) d) P(draw a red first and a blue second) P(draw a blue first and a blue second) P(both draws were neither red or green) P(1st draw was red and the second draw was not red) 2. You are dealt two cards (without replacement) from a standard deck of 52 cards. a) b) c) d) What is the probability that the first card you are dealt is an ace? What is the probability that the second card you are dealt is a Jack? What is the probability that you are dealt an Ace and a Jack in any order? If an ace is worth 11 points and a 10, Jack, Queen, and King are worth 10 points each, what is the probability that you will obtain 21 points with two cards? 3. You survey your classmates to determine their favorite flavor of ice cream. You think males may prefer vanilla and females may prefer chocolate, so you tally their responses separately. You found the following results: Males Females Vanilla 23 8 Chocolate 10 17 Other 10 9 a) What is the probability that a randomly selected student prefers chocolate? b) What is the probability that a randomly selected student is a female and prefers chocolate? c) If the student chosen is a female, what is the probability that she prefers chocolate? d) Are the events “female” and “prefers chocolate” independent or dependent? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 80 of 120 Mathematics I 1st Edition Unit 4 Homework Day 1 Sample Yahtzee Unit Name:________________________ Given the following frequency distributions: a) Draw a histogram b) Calculate the mean and mean deviation of the distribution c) Calculate the five number summary d) Based on the shape of the distribution, determine which measure of center and spread should be used. 1. X 2 F(x) 7 2. X F(x) 0 7 3 11 1 5 4 10 2 3 5 6 3 6 4 9 3. You roll one die 80 times. It lands on the number one 10 times. It lands on the number two 14 times. It lands on the number three 12 times. It lands on the number four 9 times. It lands on the number five 15 times. a) How many times did it land on 6? b) Make a frequency distribution below where X is the outcome of the roll and F(x) is how many times that outcome occurred. c) Graph the frequency distribution as a histogram below. d) Calculate the mean and mean deviation of the distribution Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 81 of 120 Mathematics I Unit 4 1st Edition e) Calculate the five number summary f) Based on the shape of the distribution, determine which measure of center and spread should be used. g) What should the shape of this distribution look like if the die was fair? Do you think that this die is fair? 4. You selected a sample of 50 girls from your school and recorded their heights (in inches…rounded to the nearest inch). The heights were: 60, 60, 61, 67, 66, 64, 64, 63, 63, 62, 69, 67, 64, 64, 63, 65, 65,65, 70, 68, 67, 64, 64, 65, 66, 66, 68, 62, 62, 63, 64, 65, 63, 63, 65, 64, 66, 67, 69, 60, 61 ,61, 63, 68, 69, 67, 67, 65, 66, 65 a) Make a frequency distribution of the girls’ heights below: b) Draw a histogram of this distribution below: c) Calculate the mean and mean deviation of the distribution: d) Calculate the 5 number summary: e) Based on the shape of the distribution, determine which measure of center and spread should be used. f) What would you expect the shape of this distribution to look like? Do you think that the sample was selected randomly? Explain. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 82 of 120 Mathematics I 1st Edition Unit 4 Homework Day 2 Sample Yahtzee Unit Name:__________________ Given the following probability distributions: a) Draw a histogram b) Calculate the mean and mean deviation of the distribution c) Calculate the five number summary d) Based on the shape of the distribution, determine which measure of center and spread should be used. 1. X 0 P(x) .25 2. X P(x) 1 .17 1 .4 2 .25 2 .3 3 .22 4 .11 3 .2 4 .1 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 83 of 120 Mathematics I 1st Edition Unit 4 You flip a coin 4 times. The coin either lands on heads or tails. Let X= Number of heads that you land on in 4 flips. a) List the sample space below. (Record all possible outcomes of four flips below such as HHHH, HHHT, etc.) X P(x) b) Fill in the probability distribution below: 0 1 2 3 4 c) Draw the probability histogram associated with the distribution in part (b) d) Calculate the mean and mean deviation of the distribution. e) Calculate the 5 number summary. f) Based on the shape of the distribution, which measure of center and spread should be used? g) Use the distribution to answer the following questions: 1. What is the probability of landing on no heads when the coin is flipped four times? 2. What is the probability of landing on at least one head when the coin is flipped four times? 3. What is the probability of landing on 2 or more heads when the coin is flipped four times? 4. What is the probability of landing on less than 3 heads when the coin is flipped four times? 5. What is the probability of landing on at least 2 tails when the coin is flipped four times? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 84 of 120 Mathematics I 1st Edition Unit 4 Yahtzee Unit Sample Homework Day 3 Name:________________________ 1. You go to a carnival and pay $5 to play a game. The probability that you win the game is 0.4. If you win, you get $7. If you lose, you get nothing. a) Fill in the probability distribution below: Outcomes Win $7 Win $0 P(Outcome) .4 b) What are your expected winnings (not taking the $5.00 into consideration) if you play the game one time? c) What is your net winnings (or loss) if you play the game one time? d) How much would you expect to win/lose if you played the game 100 times? e) Is this game fair? Explain. f) Make this game fair by changing only the amount of money that you would win. In other words, how much money would you have to win in order for you to break even over a long period of time? 2. Two players play a game with two special four sided dice. When either of these dice is rolled, each face has an equal chance of landing on top. Die A has two 5’s and two 7’s on its faces. Die B has three 6’s and one 9 on its faces. The first player selects a die and rolls it. The second player rolls the remaining die. The winner is the player whose die has the higher number on top. a) Which die is more likely to win? Why? b) If the player who uses Die A gets 25 tokens each time he or she wins the game, how many tokes must the player using die B receives in order for the game to be fair (they have the same number of tokens in the long run)? c) Suppose the players roll the same die twice. The winner is the player who has the greatest sum of the two rolls. Which die is more likely to win? Why? Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 85 of 120 Mathematics I 1st Edition Unit 4 Yahtzee Unit Sample Homework Day 4 Name:________________________ Suppose you play a game with four dice. What is the probability of getting the following on your first roll? Show all work. 1. Four of a kind 2. Only three of a kind (4 of a kind would not count) 3. Only two of a kind 4. No pairs (all different outcomes) 5. Two pairs 6. Create a game using some or all of your probabilities above. Assign points to your game based on the probabilities. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 86 of 120 Mathematics I 1st Edition Unit 4 Solutions Culminating Task #1 For Wheel of Fortune Unit…Spinning our Wheels A student created a spinner and recorded the outcomes of 100 spins in the table and bar graph below. Amount of Money on each section of the spinner Number of times that the student landed on that section $0 $100 $200 $300 $400 $500 $600 30 10 15 20 10 10 5 Student's outcomes from 100 spins 35 30 Frequency 25 20 15 10 5 0 Money per spinner section Based on the table and graph above, calculate the experimental probabilities of landing on each section of the spinner. Use these probabilities to draw what the spinner would look like below: The experimental probabilities are as follows: P($0) = 30/100 or .3, P($100)= 10/100 =.1, P($200) = .15, P($300)= .2, P($400) = .1, P($500) = .10, P($600)= .05 Calculate the average amount of money a person would expect to receive on each spin of the spinner. $200 $100 $300 $0 0(.3) + 100(.1) +200(.15) + 300(.2) + 400(.1) + 500(.1) + 600(.05) = $220 Calculate the probability that you will receive at least $400 on your first spin. .25 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 87 of 120 $400 $500 $600 Mathematics I 1st Edition Unit 4 Given you land on $300 the 1st time, what is the probability that the sum of your first two spins is at least $600? You can land on (300, 300), (300,400), (300, 500), or (300, 600). So the answer is 4/7. What is the probability that the sum of 2 spins is $400 or less? For two spins, there are 7*7 or 49 possible outcomes. If the sum is $400 or less, then you can land on (0,0), (0, 100), (0, 200), (0, 300), (0, 400), (100, 100), (100, 200), (100, 300), (200, 200) (100, 0), (200, 0), (300, 0), (400, 0), (200, 100), or (300, 100). So, the probability is 15/49 You propose a game: A person pays $350 to play. If the person lands on $0, $200, $400, or $600, then they get that amount of money and the game is over. If the person lands on $100, $300, $500, then they get to spin again, and they will receive the amount of money for the sum of the two spins. In the long run, would the player expect to win or lose money at this game? If the player played this game 100 times, how much would he/she be expected to win or lose? Event 0 P(Event) .30 Money $0 won 200 .15 $200 400 .10 $400 600 .05 $600 Event: P(event) Money won: (100, 0) (100,100) (100,200) (100,300) (100,400) (100, 500) (100, 600) (.10)(.30)=.03 (.1)(.1) = .01 (.1)(.15) = .015 (.1)(.2) = .02 (.1)(.1) = .01 (.1)(.1) = .01 (.1)(.05) = .005 $100 $200 $300 $400 $500 $600 $700 100* 300* 500* Exp. amt. money won if 100 is first: $32 Event: (300, 0) (300, 100) (300, 200) (300, 300) (300, 400) (300, 500) (300, 600) P(Event) (.2)(.3) = .06 (.2)(.1) = .02 (.2)(.15) = .03 (.2)(.2) = .04 (.2)(.1) = .02 (.2)(.1) = .02 (.2)(.05) = .01 Money Won: $300 Exp. amt. money if 300 is 1st: $104 $400 $500 $600 $700 $800 $900 (500, 0) (500, 100) (500, 200) (.1)(.3)=.03 (.1)(.1)=.01 (.1)(.15) = .015 $500 Exp. amt. money if 500 is 1st: $72 $600 $700 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 88 of 120 Mathematics I (500, 300) (500, 400) (500, 500) (500, 600) Unit 4 (.1)(.20)= .02 (.1)(.1) = .01 (.1)(.1) = .01 (.1)(.05) = .005 1st Edition $800 $900 $1000 $1100 Total Expected amount of money won: .3(0) + .15($200) + .1($400) +.05($600) + $32 + $104 + $72 = $308 Since the person payed $350 to play the game, then the person will lose about $42 each time he/she plays. If the player continues to play 100 times, then he/she is expected to lose 100($42) or $4200. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 89 of 120 Mathematics I Unit 4 1st Edition Culminating Task #2 Wheel of Fortune…Let’s Play to Win Susan played the bonus round of wheel of fortune 30 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 17, 18, 19, 21, 24, 24, 24, 26, 28, 31, 33, 34, 35, 35, 37, 40 Monique also played to bonus round of wheel of fortune 25 times. She recorded how long it took her to guess the phrase to the nearest second. The following are the lengths of time it took her to guess each phrase correctly: 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 55 a) Graph the two distributions below. Which measure of center (mean or median) is more appropriate to use and why? Calculate that measure of center. Susan’s Distribution: mean = 20.9333 median = 17.5 Monique’s Distribution: mean = 16.56 median = 15 Both distributions are skewed right; therefore, the median will be the better measure of center. Monique’s distribution has an outlier at 55. Without the outlier, the graph of the distribution will be approximately symmetric. Due to the outlier, the median is a better measure of center. b) Comment on any similarities and any differences in Susan’s and Monique’s times. Make sure that you comment on the variability of the two distributions. The IQR of Susan’s times is 28-13 = 15. The range of Susan’s times is 40-10 = 30. The IQR of Monique’s times is 16.5-14 = 2. The range of Monique’s times is 55-12 = 43. Although Monique had a larger range of scores than Susan, it was due to her outlier of 55. Most of the time, Monique was much more consistent than Susan. The middle 50% of her scores were within 2 seconds of each other. Whereas, the middle 50% of Susan’s scores were within 15 seconds of each other. c) If you are only allowed 15 seconds or less to guess the phrase correctly in order to win, which girl was more likely to win and why? 50% of Monique’s guesses were 15 seconds or less. Whereas, 50% of Susan’s guesses were 17.5 seconds or less. Therefore, Monique had a higher percentage of guesses less than or equal to 15 seconds. d) If Susan found that she could have guessed each phrase 3 seconds faster if she had chosen a different set of letters, would that have made any difference in your answer to part c? Why/why not? Yes, Susan’s median would become 17.5-3 or 14.5 seconds. Therefore, she would have at least 50% of her guesses less than 15 seconds. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 90 of 120 Mathematics I 1st Edition Unit 4 Culminating Activity #1 True/False Unit A teacher makes up a 5 question multiple choice test. Each question has 5 answers listed “a-e.” A new student takes the test on his first day of class. He has no prior knowledge of the material being tested. a) What is the probability that he makes a 100 just by guessing? If the student makes a 100%, then he answered all 5 questions correctly. The P(correct) = 1/5. Therefore, the answer is (1/5)^5 = 1/3125 b) What is the probability that he only misses 2 questions? 3 2 1 4 If he misses 2 questions, then three are correct. P(3 correct) = 5 C3 = .0512 5 5 c) What is the probability that he misses more than 2 questions? That means that he misses 3, 4, or 5 questions. Or, he gets 0, 1, or 2 correct. So, 0 5 1 4 2 3 1 4 1 4 1 4 5 C0 5 C1 5 C2 = .94208 5 5 5 5 5 5 d) Let X= the number correct on the test. Make a graphical display of the probability distribution below. Comment on its shape, center, and spread. From a histogram, you should see that the shape is skewed to the right. The mean = np = 1 (which is one measure of center). The median is also 1. The IQR = 2-0 = 2. The range = 5. The mean deviation is .65536. e) Based on your distribution, how many questions should the new student get correct just by randomly guessing? One (since that occurs the most often and is also the mean). f) The answers to the test turned out to be the following: 1. A 2. A 3. A 4. A 5. C Do you think that the teacher randomly decided under which letter the answer should be placed when she made up the test? Explain. The student will probably use simulations to support his answer. If your students understood the log formula (which is not an objective at this level), then they could have shown that the longest string of letters in a row is about 1 for a 5 question test. Therefore, if the answers were randomly generated, you would not expect to have 4 A’s in a row. 4 log 1 5 log5 (4) .86135 5 1/ 5 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 91 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #2 True or False Unit Given a standard deck of 52 cards which consists of 4 queens, 3 cards are dealt, without replacement. 7. What is the probability that all three cards are queens? (4/52)*(3/51)*(2/50) = 24/132600 8. Let the first card be the queen of hearts and the second card be the queen of diamonds. Are the two cards independent? Explain. No, because if the 1st card is not replaced, then the second card drawn will have a different probability. 9. If the first card is a queen, what is the probability that the second card will not be a queen? 48/51 10. If the first two cards are queens, what is the probability that you will be dealt three queens? 2/50 11. If two of the three cards are queens, what is the probability that the other card is not a queen? 1 12. Answer questions #1 and #2 if each card is replaced in the deck (and the deck is well shuffled) after being dealt. #1. (4/52)(4/52)(4/52) = 64/140608 #2. Each draw should be independent of each other since the cards are replaced and well shuffled. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 92 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #1 Yahtzee Unit Two players play a game. The first player rolls a pair of dice. If the sum is 6 or less, then player 1 wins. If it’s more than six, then player 2 gets to roll. If player 2 gets a sum of 6 or less, then he loses. If player 2 gets a sum greater than 6, then he wins. a) Which player, player 1 or player 2, is more likely to win? Why? Player 1 is more likely to win. The P(player 1) wins is 15/36. The P(player 2) wins = (21/36)(21/36) = 49/144. Player 2 only wins if player one does not win on the first roll and player two rolls a sum greater than 6. Player 2 could either lose on Player 1’s first roll, or player 2 could get a chance to roll and then roll a sum of 6 or less and lose. The probability that player 2 gets to roll and loses is (21/36)(15/36)=35/144. b) If player 1 is awarded 10 tokens each time he/she wins the game, how many tokens must player 2 be awarded in order for this to be a fair game? Why? You must solve the following equation in order to get the answer: 15 49 (10) ( x) So, player 2 must be awarded 12.244 tokens to be fair. If he’s awarded 12 36 144 tokens, the game is slightly in player 1’s favor. If he’s awarded 13 tokens, the game is slightly in player 2’s favor. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 93 of 120 Mathematics I Unit 4 1st Edition Culminating Activity #2 Yahtzee Unit A student rolled 3 dice 100 times, found the sums of the 3 dice, and put them into the following frequency distribution: Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Frequency 1 3 5 4 8 9 13 15 12 13 7 3 6 1 2 0 Experimental Probability .01 .03 .05 .04 .08 .09 .13 .15 .12 .13 .07 .03 .06 .01 .02 0 a) Based on the student’s simulation, compute the experimental probabilities for the sum of 3 dice and write them in the table above. See the table above. b) Based on the student’s simulation, what is the expected value (the mean) of the sum of the three dice? 3(.01) + 4(.03) + …..18(0) = 10 c) Based on the student’s simulation, what is the median sum of the three dice? The median is also 10. d) Comment on the relationship between the mean and median relative to the shape of the distribution. The distribution is approximately symmetric. Therefore, the mean should be close to the median. In this case, they are the same. e) Based on the student’s simulation, what is the probability that the sum of 3 dice is even? .03 + .04 + …. + 0 = .48 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 94 of 120 Mathematics I Sum of Three Dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Unit 4 Theoretical probability P(x) 1/216 3/216 6/216 10/216 15/216 21/216 25/216 27/216 27/216 25/216 21/216 15/216 10/216 6/216 3/216 1/216 f) 1st Edition Based on the theoretical probabilities in the table to the left, what is the expected value (the mean) of the sum of the three dice? 3(1/216) + 4(3/216) + …. + 18(1/216) = 10.5 g) Based on the theoretical probabilities, what is the median sum of the three dice? The median is also 10.5. h) Comment on the relationship between the mean and median relative to the shape of the distribution. The theoretical distribution is perfectly symmetric (bell-shaped). Therefore, the mean = median. i) Based on the theoretical probabilities, what is the probability that the sum of 3 dice is even? 3/216 + 10/216 + … + 1/216 = .5 j) How does the theoretical probability that the sum of 3 dice is even compare to the experimental probability that the sum of 3 dice is even (part e). They are pretty close. The experimental probability was .48 for only 100 rolls. The theoretical is .5. k) How does the theoretical mean and median compare to the experimental mean and median from the student’s simulation? Again, they are pretty close. The theoretical mean and median is 10.5; whereas, the experimental mean and median is 10. l) Display the experimental probability distribution and the theoretical probability distribution graphically so that they can be easily compared. The student should draw side to side boxplots or histograms. If you look at the boxplots, the experimental distribution has a smaller IQR than the theoretical distribution. The median is also slightly smaller for the experimental distribution. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 95 of 120 Mathematics I Unit 4 1st Edition m) Based on your answers to parts “j, k, and l” above, do you think that the student really simulated rolling 3 dice 100 times, or did the student make up the data. Explain. The student’s explanation should include a direct comparison to the experimental vs. theoretical summary statistics. He/she should also use the concept of “randomness” in the explanation. An argument for a student who believes that the student did not fabricate the data might include: It would be unrealistic to get a perfectly symmetric distribution from the experimental data for only 100 rolls. But, a person should expect a somewhat symmetric distribution from rolling 3 dice. If it was perfectly symmetric or strongly skewed, it would be unlikely that the distribution was generated from the random event of rolling 3 dice. An argument for a student who believes that the data was fabricated might include: Since the mean is exactly the same as the median for the experimental data, then it is unlikely to be generated by a rolling 3 dice. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 96 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit…Solutions Sample Homework Day 1 Name:__________________ 1. The large circle has radius of 7 units. Each of the four smaller circles have radius of 1 unit. A point is chosen at random from inside the circle. What is the probability that the point will not be in one of the four smaller circles? 49/45 2. A square is formed by the intersection of the lines x=2, x=-1, y=-1, and y=2. A point is randomly chosen from inside the square. c) What is the probability that the point lies above the line y=x? 1/2 d) What is the probability that the point lies above the line y 2 2 x ? 1/3 3 3 3. Given the spinner: 100 400 200 300 f) g) h) i) j) What is the probability of landing on 400? 1/4 What is the probability of landing on a number less than 300? 1/2 What is the probability of landing on a number greater than 100? 3/4 If the spinner is spun twice, what is the probability that the sum is 500? 1/4 If the spinner is spun twice, what is the probability that the sum is less than 500? 3/8 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 97 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit…Solutions Sample Homework Day 2 Name:________________________ Assume all events described in the problems below are independent. 1. Sam makes 70% of his free throws. What is the probability that a) Sam makes 2 free throws in a row? .49 b) Sam makes his first free throw and then misses his next free throw? .21 c) Sam misses his first free throw and then makes his next? .21 d) Sam misses both free throws? .09 e) Sam makes at least one free throw? .91 2. If you roll a die, the probability that you land on a “one” is 1/6. If you roll the same die twice, what is the probability that: a) You land on two “ones” in a row? 1/36 b) You never land on a “one”? 25/36 c) You land on at least one “one”? 11/36 3. A sack contains 7 blue, 3 red, and 2 green marbles. If you draw a marble, replace it, and then draw another marble, what is the probability that: a) You draw a red marble and then a green marble? 1/24 b) You draw a red marble and then another red marble? 1/16 c) You draw at least one blue marble? 119/144 d) You draw at least one red or blue marble? 25/36 e) You never draw a blue marble? 25/144 f) Which is more likely to occur and why? To draw the marble colors RBBR or to draw the marble colors BBGRBB? RBBR because P(RBBR) = .02127 and P(BBGRBB) = .004824 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 98 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit… Solutions Sample Homework Day 3 Name:________________________ 1. Given the probability distribution, calculate the expected value (the mean). 1.3 x P(x) 0 .3 1 .2 2 .4 3 .1 2. Suppose the probability of winning a game is 1/100. In order to play the game, you must pay $5. If you win, you will receive $200. Outcome P(outcome) Payoff Win 1/100 $200 Lose 99/100 $0 a) Fill in the probability distribution above. b) How much money do you expect to win each time you play the game (do not take into account the $5.00). $2 c) What is your “net” loss each time that you play? Lose $3 each time you play d) If you played the game 50 times, how much money would you win/lose? $150 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 99 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit… Solutions Sample Day 5 homework Name:__________________ Note: The homework for day 4 should be to work on the project. 1. How many ways are there to rearrange all of the letters a) To the word “cat?” 3! = 6 b) To the word “cat” if “c” must be the first letter of each arrangement? 2 2. How many ways are there to rearrange all of the letters a) To the word “book”? 12 b) To the word “book” if “k” must be the first letter of each arrangement? 3 c) To the word “book” if “o” must be the first letter of each arrangement? 6 3. How many ways to rearrange all of the letters a) To the word “Mississippi?”34650 b) To the word “Mississippi” if “M” must be the first letter and “P” must be the last letter of each arrangement? 630 4. If each of four students shakes hands with each of the other 3 students one time, how many handshakes were made? 6 5. A locker has a combination lock with 40 numbers on it. Three numbers must be selected in the exact order or the lock will not open. How many combinations are possible if numbers can be repeated? How many combinations are possible if numbers cannot be repeated? 64000, 59280 6. Six people are seated at a circular table for dinner. The hostess tries to determine the best way to seat everyone. How many different ways can she seat these six people? 120 If there are 3 men and 3 women, How many different ways can she seat these six people if she wants to have men sitting next to only women? 3!2!=12 7. An addition was made to an office building which provided 6 new offices. Currently, there are 10 staff members of equal ability who need offices. c) How many ways can the office manager assign the 10 staff members to the 6 new offices (obviously, 4 staff members will still not have offices)? 151200 d) There are 5 male staff members and 5 female staff members. The office manager decides that 3 males and 3 females must be placed in the 6 offices. How many ways can this be done? 3600 8. A true/false test has 10 questions. Five answers are “true” and five answers are “false.” How many ways can you rearrange the answers “TTTTTFFFFF”? 252 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 100 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit… Solutions Sample Homework Day 6 Name:________________________ 1. A class had the following test averages on their 8 unit tests: 93, 82, 91, 64, 78, 90, 86, 87 a) b) c) d) e) f) g) h) Write the 5 number summary below: 64, 80, 86.5, 90.5, 93 Compute the interquartile (IQR) range: 10.5 What percent of data is contained in the IQR? 50% Draw the boxplot. See graph Describe the shape of the boxplot. Skewed left with an outlier at 64 Compute the mean of the class test averages. 83.875 Compute the mean deviation of the class averages. 6.90625 Which measure of center is more appropriate to use and why? Median since the distribution is skewed i) 25% of scores are at or below 80 j) 25% of scores are at or above 90.5 k) 50% of scores are between 64 and 86.5 or 80 and 90.5 or 86.5 and 93. 2. A student had 5 homework grades: 68, 99, 72, 100, and another one that he lost. The teacher told him that his homework average is 82. What was the 5th homework grade? 71 3. A student has taken 4 tests. Her test average is 82. What must the score be on her next test if she wants to raise her average to 83? 87 4. Given the numbers the consecutive integers 1, 2, 3,…., 31. What is the mean? What is the median? Both are 16 5. If the median of a list of 31 consecutive integers is 42, what are the least and greatest integers in the list? 27 and 57 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 101 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit… Solutions Sample Homework Day 7 Name:__________________ Sue and John had the following test scores: Sue’s scores: 79, 77, 78, 80, 80, 81, 83, 82, 80 John’s scores: 65, 60, 68, 80, 81, 92, 100, 95, 79 1. Calculate Sue’s five number summary: 77, 78.5, 80, 81.5, 83 2. Calculate John’s five number summary: 60, 66.5, 80, 93.5, 100 3. Draw Sue’s and John’s boxplots next to each other. See student graphs 4. Compare their distributions. Both graphs have the same median, but Sue has little variance in her scores. John’s scores vary greatly compared to Sue’s. Both distributions appear to be symmetric. 5. Calculate Sue’s mean and mean deviation: Sue’s mean is 80, and her mean deviation is 1.33333. 6. Calculate John’s mean and mean deviation: John’s mean is also 80. But his mean deviation is 10.6666. 7. Compare Sue’s and John’s averages and variability. While their averages are the same, John’s scores are more variable than Sue’s scores. 8. Which measure of center is more appropriate to use? Why? The mean is fine to use since the distributions are both symmetric. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 102 of 120 Mathematics I 1st Edition Unit 4 Wheel of Fortune Unit…. Solutions Sample Homework Day 8 Name:________________________ 1. Janet and Ed had the following scores: Janet’s scores: 50, 90, 70, 75, 80, 80, 90, 85, 90, 90 Ed’s scores: 75, 75, 80, 90, 95, 80, 80, 75, 75, 75 Compare Janet’s and Ed’s scores. Make sure that you display the data graphically. Calculate the summary statistics and use them when you discuss the similarities and differences in the distributions’ centers, shapes, and variabilities. Janet’s distribution is skewed to the left with an outlier of 50. Ed’s distribution is somewhat symmetric if you ignore the two outliers of 90 and 95 (skewed right if you do not). Janet has much more variability in her scores than Ed. Her IQR is15, and Ed’s IQR is only 5. Janet’s median is higher than Ed’s. Her median score is 82.5; whereas, Ed’s median score is only 77.5. Although Janet had the lowest score (50), in general, she outperforms Ed. Less than 25% of Ed’s scores are above Janet’s median. Both Janet and Ed have 75% of their scores at or above 75. 2. If 3 points were added to each of Janet’s scores, how would the following statistics change? a) b) c) d) e) f) g) Mean: From an 80 to an 83 Median: From 82.5 to 85.5 Range: Stays the same IQR: Stays the same Mean Deviation: Stays the same Q1 (lower quartile): From 75 to 78 Q3 (upper quartile): From 90 to 93 3. If each of Ed’s scores were multiplied by 1.1, how would the following statistics change? a) Mean: From 80 to 88 b) Median: From 77.5 to 85.25 c) Range: 20 to 22 d) IQR: From 5 to 5.5 e) Mean Deviation: From 5 to 5.5 f) Q1 (lower quartile): From 75 to 82.5 g) Q3 (upper quartile): From 80 to 88 4. Give a list of 5 numbers that are not equal to zero such that the mean deviation of those 5 numbers is equal to zero. 7, 7, 7, 7, 7 (all numbers must be the same) Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 103 of 120 Mathematics I True or False ... Solutions Sample Homework Day 1 1st Edition Unit 4 Name:________________________ 1. How many ways are there to rearrange the letters ABBBAAAB? 70 2. How many different ways are there for a coin that is flipped 8 times to land on 3 tails and 5 heads? 56 3. How many different ways are there to choose a committee of 3 students from 10 students? 120 4. How many different ways are there to choose a president, vice-president, and secretary from 10 students (if a student cannot hold more than one office)? 720 5. What is the difference between question #3 and #4? A committee is a group where order is not important. Therefore, combinations are appropriate. In problem #4, order is important; therefore, permutations are appropriate. 6. If you roll 3 dice, how many ways can exactly one of the dice land on 6? 60+15 = 75 (there are 60 different ways if all 3 dice are different, but one must be a 6….there are 15 ways if the other two dice are the same) 7. If you roll 3 dice, how many ways can exactly two of the dice land on 6? 15 8. Write a question that is solved by the formula 5 P2 . See student work. 9. Write a question that is solved by the formula 5 C2 . See student work. 10. How many different hands of 5 cards are there in a deck of 52 cards? 2598960 11. A bag contains 6 yellow marbles, 3 blue marbles and 8 red marbles. How many ways are there to choose 3 yellow, 1 blue, and 2 red marbles? 1680 12. If you randomly select 6 marbles from the bag, what is the probability that you select 3 yellow, 1 blue, and 2 red marbles (if the bag contains 6 yellow, 3 blue, and 8 red marbles)? 30/221 = .13575 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 104 of 120 Mathematics I Unit 4 True or False Unit … Solutions Sample Homework Day 2 1st Edition Name:__________________ 1. Suppose you guess on a multiple choice test where each question has 4 answers. a) What is the probability of guessing correctly? 1/4 b) If there are 5 questions, what is the probability that you get all questions correct by guessing? 1/1024 c) If there are 5 questions, what is the probability that you get 4 or less correct by guessing? 1023/1024 d) If there are 5 questions, what is the probability that you get 2 incorrect? 45/512 e) If there are 5 questions, what is the probability that you get at most 2 incorrect? 53/512 2. Matthew is a 70% free throw shooter. Let x = # shots that Matthew makes. Let p(x) represent the probability of making that many shots. Matthew shoots 6 free throws. g) Fill in the probability distribution below: X 0 1 2 3 4 5 6 P(x) .000729 .010206 .059535 .18522 .324135 .302526 .117649 a) Draw a histogram of the probability distribution below: see student work…it should appear skewed left b) Describe the shape, center and spread of the distribution: skewed left….mean = 4.2 and median = 4…..mean deviation = .907578 and IQR - 2 c) Using your distribution in (a) or (b), what is the probability that Matthew makes at least 4 free throws? .74433 d) What is the probability that he misses at least 4 free throws? .07047 e) How many free throws do you expect Matthew to make? About 4 3. Suppose a rocket is designed to have 50 important parts. If any one of these parts fail, then the rocket will crash. Therefore, each part was designed to be 99% reliable. Assuming that each part operates independently, then what is the probability that a) All of the parts work and the rocket does not crash? .605 b) Only one of the parts does not work? .30556 c) How reliable must the parts be in order to insure that there is a 95% probability of all 50 parts working? Each part must be 99.8975% reliable Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 105 of 120 Mathematics I 1st Edition Unit 4 True or False Unit… Solutions Sample Homework Day 3 Name:__________________ 1. Suppose a child born to a certain pair of parents has probability of .75 of having blood type O. These parents have 6 children. a) Let X= the number of children who inherit blood type O. Fill in the probability distribution below: X 0 1 2 3 4 5 6 P(x) .00024 .00439 .03296 .13184 .29663 .35596 .17798 b) Draw a histogram of the probability distribution below: see student work c) Describe the shape, center and spread of the distribution: skewed left….mean = 4.5 and median =5…..mean deviation = 1.585 and IQR = 1 d) Using your distribution in (a) or (b), what is the probability that at least 3 children will inherit blood type O? .9624 e) What is the probability that fewer than two children will not inherit blood type O? .5339 f) How many children do you expect to inherit blood type O? 4.5 2. Suppose a blood test given to a woman when she is 12 weeks pregnant is 90% accurate in detecting Downs Syndrome. a) Suppose 10 women take the test when they are 12 weeks pregnant. What is the probability that all 10 tests give an accurate reading? .3487 b) What is the probability that at least 9 of the 10 tests give an accurate reading? .736 c) What is the probability that at least 7 of the 10 tests are accurate? .9872 d) Let X= the number of accurate tests. Draw a histogram of the probability distribution below. See work e) Describe the center, shape, and spread of the distribution. Skewed left….mean = 9 and median = 9….mean deviation = 4.003 and IQR = 2 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 106 of 120 Mathematics I Unit 4 True or False Unit… Solutions Sample Homework Day 5 1st Edition Name:__________________ 1. You survey a random group of students to find out which subject they like the most. You organized their responses into the table below: Math Science Language Arts Social Studies Freshmen 21 18 32 29 Sophomores 33 35 21 19 Juniors 25 28 22 32 Seniors 28 25 21 19 a) How many students participated in the survey? 408 b) What is the probability that a randomly selected student liked math the most? 107/408 c) What is the probability that a randomly selected student liked science the most? 106/408 d) What is the probability that a randomly selected student liked Language Arts the most? 96/408 e) What is the probability that a randomly selected student liked Social Studies the most? 99/408 f) Of the freshmen, what is the probability that their favorite subject was either math or science? 39/100 g) Of the sophomores, what is the probability that their favorite subject was neither math or science? 40/108 h) Of the juniors, what is the probability that their favorite subject was either math or social studies? 57/107 i) Of the seniors, what is the probability that their favorite subject was neither math or social studies? 46/93 2. A student knows about 25% of the material. The student takes a 10 question multiple choice test. Each question has 4 answers. a) What percent of the questions do you expect the student to answer correctly? 43.75% b) If the student answers correctly, what is the probability that she guessed? .42857 3. A company car was involved in a hit and run accident at night. 70% of the company cars are green and 30% of the company cars are blue. A witness at the scene identified the car in the accident as blue. The witness was tested under similar visibility conditions and correctly identified colors 80% of the time. What is the probability that the company car involved in the accident was blue? .38 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 107 of 120 Mathematics I 1st Edition Unit 4 True or False Unit… Solutions Sample Homework Day 6 Name:__________________ 1. Choose a point at random in the rectangle with boundaries 1 x 1 and 0 y 3 . This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x-coordinate and Y be the y-coordinate of that point chosen. Find the following, and justify all answers numerically and graphically (or both): a) b) c) d) P(Y 1 and X 0) 1/6 P(Y 2 or X 0) 2/3 P (Y X ) 11/12 P(Y 2 Y X ) 7/11 2. A point is randomly selected in the large 8 x 8 square below. Within the 8 x 8 square, there is a circle with radius 2. a) Find the probability that the point selected is inside the circle. (4pi)/64 b) Find the probability that the point selected is inside the smaller square. 1/8 c) Given the point is inside the circle, what is the probability that it is inside the smaller square? 8/(4pi) = .63662 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 108 of 120 Mathematics I 1st Edition Unit 4 True or False Unit… Solutions Sample Homework Day 7 Name:__________________ 1. A bag contains 3 red marbles, 5 green marbles, and 2 blue marbles. Two consecutive draws are made from the bag without replacement of the first draw. Find the probability of each event: a) b) c) d) P(draw a red first and a blue second) 6/90 P(draw a blue first and a blue second) 2/90 P(both draws were neither red or green) 2/90 P(1st draw was red and the second draw was not red) 21/90 2. You are dealt two cards (without replacement) from a standard deck of 52 cards. a) What is the probability that the first card you are dealt is an ace? 4/52 b) What is the probability that the second card you are dealt is a Jack? 4/51 c) What is the probability that you are dealt an Ace and a Jack in any order? 2*(4/52 * 4/51) = 8/663 d) If an ace is worth 11 points and a 10, Jack, Queen, and King are worth 10 points each, what is the probability that you will obtain 21 points with two cards? 3*(8/663) = 24/663 3. You survey your classmates to determine their favorite flavor of ice cream. You think males may prefer vanilla and females may prefer chocolate, so you tally their responses separately. You found the following results: Males Females Vanilla 23 8 Chocolate 10 17 Other 10 9 a) What is the probability that a randomly selected student prefers chocolate? 27/77 b) What is the probability that a randomly selected student is a female and prefers chocolate? 17/77 c) If the student chosen is a female, what is the probability that she prefers chocolate? 17/34 d) Are the events “female” and “prefers chocolate” independent or dependent? Dependent Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 109 of 120 Mathematics I 1st Edition Unit 4 Homework Day 1… Solutions Sample Yahtzee Unit Name:________________________ Given the following frequency distributions: a) Draw a histogram b) Calculate the mean and mean deviation of the distribution c) Calculate the five number summary d) Based on the shape of the distribution, determine which measure of center and spread should be used. 1. X 2 F(x) 7 3 11 4 10 5 6 Mean = 3.44 mean deviation = 1 5 # summary: 2, 3, 3, 4, 5 It’s a little skewed to the right, so the median would be a better measure of center. 2. X F(x) 0 7 1 5 2 3 3 6 4 9 Mean = 2.16666 mean deviation = 1.2 5 # summary: 0, 1, 2.5, 4, 4 It’s approximately symmetric with two peaks at each end. The median would probably be a little better. 3. You roll one die 80 times. It lands on the number one 10 times. It lands on the number two 14 times. It lands on the number three 12 times. It lands on the number four 9 times. It lands on the number five 15 times. a) How many times did it land on 6? 20 b) Make a frequency distribution below where X is the outcome of the roll and F(x) is how many times that outcome occurred. See work c) Graph the frequency distribution as a histogram below. See work d) Calculate the mean and mean deviation of the distribution mean = 3.8125 mean deviation = 1.5 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 110 of 120 Mathematics I Unit 4 1st Edition e) Calculate the five number summary 1, 2, 4, 5.5, 6 f) Based on the shape of the distribution, determine which measure of center and spread should be used. It’s not symmetric, so the median would be a better measure of center g) What should the shape of this distribution look like if the die was fair? Do you think that this die is fair? The shape should be uniform…each outcome should be just as likely. Possibly. However, there seems to have been too many 6’s. 5. You selected a sample of 50 girls from your school and recorded their heights (in inches…rounded to the nearest inch). The heights were: 60, 60, 61, 67, 66, 64, 64, 63, 63, 62, 69, 67, 64, 64, 63, 65, 65,65, 70, 68, 67, 64, 64, 65, 66, 66, 68, 62, 62, 63, 64, 65, 63, 63, 65, 64, 66, 67, 69, 60, 61 ,61, 63, 68, 69, 67, 67, 65, 66, 65 a) Make a frequency distribution of the girls’ heights below: see work b) Draw a histogram of this distribution below: see work c) Calculate the mean and mean deviation of the distribution: Mean = 64.7 mean deviation = 2.032 d) Calculate the 5 number summary: 60, 63, 65, 67, 70 e) Based on the shape of the distribution, determine which measure of center and spread should be used. It’s slightly skewed to the right so the median would be a little better. However, the mean and median are pretty close. f) What would you expect the shape of this distribution to look like? Do you think that the sample was selected randomly? Explain. I would expect the shape to be symmetric (bell-shaped). It appears that there are more short girls than expected. It may not have been a random sample. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 111 of 120 Mathematics I 1st Edition Unit 4 Yahtzee Unit.,,,Solutions Sample Homework Day 2 Name:__________________ Given the following probability distributions: a) Draw a histogram b) Calculate the mean and mean deviation of the distribution c) Calculate the five number summary d) Based on the shape of the distribution, determine which measure of center and spread should be used. 1. X 0 P(x) .25 1 .17 2 .25 3 .22 4 .11 Mean = 1.77 mean deviation = 1.1468 5 # summary = 0, .5, 2, 3, 4 The shape is almost uniform. The median will probably be a little more accurate since it’s not symmetric. 2. X P(x) 1 .4 2 .3 3 .2 4 .1 Mean = 2 Mean deviation = 0.8 5# summary: 1,1, 2, 3, 4 The shape is strongly skewed right. Therefore, the median would be a better measure of center. You flip a coin 4 times. The coin either lands on heads or tails. Let X= Number of heads that you land on in 4 flips. a) List the sample space below. (Record all possible outcomes of four flips below such as HHHH, HHHT, etc.) There should be 2^4 = 16 outcomes listed. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 112 of 120 Mathematics I X P(x) 1st Edition Unit 4 b) Fill in the probability distribution below: 0 1 2 1/16 4/16 6/16 3 4/16 4 1/16 c) Draw the probability histogram associated with the distribution in part (b) see work d) Calculate the mean and mean deviation of the distribution. Mean = 2 mean deviation = .75 e) Calculate the 5 number summary. 5 # summary: 0, 1, 2, 3, 4 f) Based on the shape of the distribution, which measure of center and spread should be used? Mean since it’s symmetric. g) Use the distribution to answer the following questions: 6. What is the probability of landing on no heads when the coin is flipped four times? 1/16 7. What is the probability of landing on at least one head when the coin is flipped four times? 15/16 8. What is the probability of landing on 2 or more heads when the coin is flipped four times? 11/16 9. What is the probability of landing on less than 3 heads when the coin is flipped four times? 11/16 10. What is the probability of landing on at least 2 tails when the coin is flipped four times? 11/16 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 113 of 120 Mathematics I 1st Edition Unit 4 Homework Day 3… Solutions Yahtzee Unit Name:________________________ 1. You go to a carnival and pay $5 to play a game. The probability that you win the game is 0.4. If you win, you get $7. If you lose, you get nothing. a) Fill in the probability distribution below: Outcomes Win $7 Win $0 P(Outcome) .4 .6 b) What are your expected winnings (not taking the $5.00 into consideration) if you play the game one time? .4(7) + .6(0) = $2.80 c) What is your net winnings (or loss) if you play the game one time? $2.80-$5.00 = -$2.20 d) How much would you expect to win/lose if you played the game 100 times? 100*(-$2.20) = -$220 e) Is this game fair? Explain. No. If it were fair, the your expected net winnings would be $0. f) Make this game fair by changing only the amount of money that you would win. In other words, how much money would you have to win in order for you to break even over a long period of time? Solve the following equation for x: .4x + .6(0) = 5 x= $12.50 2. Two players play a game with two special four sided dice. When either of these dice is rolled, each face has an equal chance of landing on top. Die A has two 5’s and two 7’s on its faces. Die B has three 6’s and one 9 on its faces. The first player selects a die and rolls it. The second player rolls the remaining die. The winner is the player whose die has the higher number on top. a) Which die is more likely to win? Why? Die B….The probability of B winning is 10/16 or 5/8….the probability of A winning is 6/16 or 3/8. b) If the player who uses Die A gets 25 tokens each time he or she wins the game, how many tokes must the player using die B receives in order for the game to be fair (they have the same number of tokens in the long run)? 15 c) Suppose the players roll the same die twice. The winner is the player who has the greatest sum of the two rolls. Which die is more likely to win? Why? Die B…The expected sum for Die A is 10(1/4) + 12(1/2) + 14(1/4) = 12. The expected sum for Die B is 12(9/16) + 15(6/16) + 18(1/16) = 13.5 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 114 of 120 Mathematics I 1st Edition Unit 4 Yahtzee Unit … Solutions Sample Homework Day 4 Name:________________________ Suppose you play a game with four dice. What is the probability of getting the following on your first roll? Show all work. 1. Four of a kind 6*(1/6)^4 = .00463 2. Only three of a kind (4 of a kind would not count) 120/1296 3. Only two of a kind 720/1296 4. No pairs (all different outcomes) 360/1296 5. Two pairs 90/1296 6. Create a game using some or all of your probabilities above. Assign points to your game based on the probabilities. See student work Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 115 of 120 Mathematics I Unit 4 1st Edition Resources for Units Websites: 1. Adjustable Spinner Game Suggestions http://www.shodor.org/interactivate/lessons/ProbabilityGeometry/ 2. A statistical study on the letters of the alphabet http://www.col-ed.org/cur/math/math48.txt 3. Yahtzee http://mathworld.wolfram.com/Yahtzee.html 4. The Yahtzee! Page http://www.yahtzee.org.uk/ 5. Three-dice Game…A Student-invented Casino Game Http://www.herkimershideaway.org/writings/dice3.htm 6. Three Cube Roll Race http://www.ciese.org/ciesemath/rolls.jpg 7. Collegeboard…AP Statistics Free Response exam questions http://apcentral.collegeboard.com/apc/members/exam/exam_questions/8357.html Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 116 of 120 Mathematics I Unit 4 1st Edition Spinner Master In order to make spinners, copy the following onto cardstock. You can use a safety pin or a paper clip as the arrow. A brad should hold it in place. I used Geometer’s Sketchpad to create the spinners. If you have access to it, you can do the following to create your own spinners. Click on the circle tool and drag Highlight only the center and the point on the circle. Go to “construct” and choose segment. Make sure the segment is highlighted, then go to “transform” and choose rotate. Type in the angle by which you wish to rotate your segment. Continue to choose “transform” and rotate by the same or different angles until your spinner is complete. Also, cuisenaire makes a transparent spinner. If you are interested, the website is www.etacuisenaire.com. Students can also use a compass and protractor to construct the circles and angles if GSP is not available. Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 117 of 120 Mathematics I 1st Edition Unit 4 $300 $600 $500 $100 $800 $700 $200 $400 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 118 of 120 Mathematics I 1st Edition Unit 4 $600 $400 $100 $300 $800 $700 $500 $200 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 119 of 120 Mathematics I 1st Edition Unit 4 $400 $100 $800 $500 Bankrupt $200 $300 $600 Georgia Department of Education Kathy Cox, State Superintendent of Schools March 14, 2008 Copyright 2008 © All Rights Reserved Unit 4: Page 120 of 120
