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Goal 1: To help practicing teachers get a fundamental understanding of t-tests and p-value in a short amount of time and apply it to biology data they obtain, without “too much math.” 2-sample t-tests p-value approach Sampling distributions & CLT Continuous probability distribution Probability Fundamentals Goal 2: Learn to use TI-84 Graphing Calculator to do statistical analysis and find p-values and understand what the results mean. AIMS – statistics workshop Day 1 page 1 Goal for Day 1: Develop an intuitive understanding of Probability without giving a “lesson” on rules of probability and don’t make it seem like math; Walk through some ideas/activities for teaching probability; Look at probability items on the ISAT; Look at probability in Connected Math; Develop an understanding of probability distributions. AIMS – statistics workshop Day 1 page 2 YOU DECIDE: Is Probability involved in the solution? 1. You buy two identical notebooks at different stores. One notebook cost $1.89, the other $.99. You started out with $11.00. How much money do you have left? 2. You set our VCR to record a TV show at 7:30 p.m. on Channel 2. You come home at 9:30 pm. Do you know what is recorded on the tape? 3. Your new computer has just arrived. You open the box, set up the computer, and flip the switch to "on." Will it work? Moral: If outcomes are uncertain -> it's probability AIMS – statistics workshop Day 1 page 3 Probability theory had its origin in the 16th Century, when an Italian physician and mathematician named Jerome Cardan wrote the first book on the subject, The Book on Games of Chance. For many years the "mathematics of chance" was used primarily to solve gambling problems. It has come a long way since then. Today, the theory of probability is, according to some mathematicians, a "cornerstone of all the sciences." People use probability to predict sales, plan political campaigns, determine insurance premiums and much more! Founders: Blaise Pascal (1623-1662) Pierre de Fermat (1601-1665) AIMS – statistics workshop Day 1 page 4 Classic probability problems: Birthday Paradox Monty's Dilema Cereal Box Problem Small World Problem Real-world applications of probability: Relative Risk Search and Rescue Queuing Theory Weather Forecasting Car and Life Insurance Lotteries and Gambling AIMS – statistics workshop Day 1 page 5 Probabilities are RATIOS, expressed as fractions decimals, or percents, determined by considering results or outcomes of experiments. Experiment = an activity whose results are determined by chance such as tossing a coin, rolling dice, drawing marbles from a bag Outcome = a result of an experiment such as tossing a head, rolling a 2, or drawing a red marble Sample Space = the list (set) of all outcomes such as H or T, {1,2,3,4,5,6}, or {Red, Blue, White} marbles Event = a subset of a sample space AIMS – statistics workshop Day 1 page 6 Probability Classical Rule or Theoretical Rule: # of times A occurs in sample space P(A) = total # of events in sample space A B 1 2 3 4 5 6 1 2 3 4 5 6 C 1 2 3 4 5 6 AIMS – statistics workshop Day 1 page 7 Empirical Rule or Experimental Rule: # of times A is observed P(A) = total # of trials A B 1 2 3 4 5 6 1 2 3 4 5 6 C 1 2 3 4 5 6 AIMS – statistics workshop Day 1 page 8 Subjective Rule: # of times A is believed to occur P(A) = total # of trials B A C 1 2 3 4 5 6 AIMS – statistics workshop Day 1 page 9 Five Great Activities Using a Spinner In the circle, which cell will the spinner most likely land on most of the time? A 1 2 3 4 5 6 C 1 2 3 4 5 6 B 1 2 3 4 5 6 D 1 2 3 4 5 6 AIMS – statistics workshop Day 1 page 10 For this experiment, you will need the spinners below. First, have one team member to use spinner P, another to use spinner R, and the third to record the results. The two players simultaneously flick their spinners and the spinner landing on the higher number wins. Each pair of players should do 25 trials. After pooling the results from the class, which spinner wins most of the time? Spinner P Spinner R 20 60 70 10 50 90 AIMS – statistics workshop Day 1 page 11 Spinner S 30 40 80 Next, have the third member take spinner S, and play against the second player using spinner R. Spin 25 times. After pooling the results from the class, which spinner wins most of the time? Finally, have the first member use spinner P play against the third member using spinner S. Spin 25 times. After pooling the results from the class, which spinner wins most of the time? AIMS – statistics workshop Day 1 page 12 Mathematical analysis: Calculate the probability of P beating R by making a table of equally likely outcomes. Spinner R 10 50 90 20 P R R Spinner P 60 P P R 70 P P R Make similar tables for R and S, and for S and P and summarize the findings. Spinner S Spinner R Spinner P Spinner S AIMS – statistics workshop Day 1 page 13 In Algebra, there is a transitive property. Ex: 4/12 = 2/6 = 1/3, then 4/12 = 1/3 In Logic, there is a reasoning argument called the chain rule (also known as hypothetical syllogism). ex: If I save, I will retire early. If I retire early, I will become lazy. Therefore, If I save, I will become lazy. In general, "if p then q" and "if q then r" are true, then "if p then r" is true. Do the spinners illustrate the transitive property or the chain rule? Do you have any idea why the spinners are named P, R, and S? AIMS – statistics workshop Day 1 page 14 From the ISAT Math Materials, grade 3 1. There are three red, two blue and two green erasers in a bag. What are the chances of picking a green eraser out of the bag without looking? a. one out of seven chances b. two out of seven chances c. three out of seven chances d. four out of seven chances 2. Amber has a box of 7 red, 22 green, 10 yellow and 2 blue paper clips. If she picked one out of the box, what color will it most likely be? a. blue b. green c. red d. yellow AIMS – statistics workshop Day 1 page 15 From the ISAT Math Materials, grade 5 1. Ellis has a box of 8 art pencils: 1 red, 3 blue, and 4 yellow. What is the probability of pulling a red pencil from the box? a. 1/8 b. 1/3 c. 3/8 d. 8/3 e. 3/1 2. Benjamin has five playing cards. They are: Black 3 Red 6 Red 4 Red 7 Black 6 If a card is drawn without looking, what is the probability that it will be an even numbered red card? a. 2/3 b. 1/5 c. 2/5 d. 3/5 e. 4/5 AIMS – statistics workshop Day 1 page 16 From the ISAT Math Materials, grade 5 3. A table shows the result of 100 spins. The spinner has 6 equal sections. What did the spinner probably look like? Color Number of times this color was spun Orange 32 White 16 Gray 36 Tan 18 AIMS – statistics workshop Day 1 page 17 The Pesky Magician A pesky Magician comes to the fair each year. He has card tricks and dice games and enjoys surprising people so that they are in awe of his powers. He seems only slightly sinister . . . almost likable . . . and it is unnerving that he uses dice and cards and things you thought you understood, but he always seems to win! You want to know more. You think there must be some trick to what he is doing, because in all your years, you’ve never seen someone repeatedly win at a game, unless there was a trick or it was “rigged.” You watch him carefully . . . you’re planning to challenge him soon! You’ve thought up a good game, so you approach the Magician and say: "I have a new game I would like to propose to you. Let's roll 2 dice and ADD them. If the sum is even, you win. If the sum is odd, I win. Your present him with a list of the possible sums: 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 AIMS – statistics workshop Day 1 page 18 But now that you see the list of possibilities, you panic! "Oh no! What did I get myself into?" you ask. Of course, the Magician likes what he sees because he seems to have the advantage. He thinks . . . hmmmm . . . 6 are even and only 5 are odd. It looks like an even answer has the advantage . . . true? A way to look at the sample space when rolling 2 dice: 1 2 3 4 5 6 7 8 9 10 11 12 AIMS – statistics workshop Day 1 page 19 The pesky Magician proposes another game to play. He says: "Here is a new game you'll like. You will roll two dice. If we subtract the smaller amount from the larger amount, the possible answers are: 0-1-2-3-4-5 I'll take the answers 0 - 1 - 2 and you take 4 - 5 - 6. OK? We tally the results. After 10 rolls, we see who wins. The Magician likes to suggest games that are in his favor. You tell him that you are not interested in playing that game, because it is strongly in his favor. "Oh, but I really want to play a game with you," he says. "How about if I can show you a diagram for a game so you can see all the possible combinations? If I can do that, will you play?" AIMS – statistics workshop Day 1 page 20 So the Magician explains, "In this game, you roll two dice and multiply them. If the answer is an even number, I get a point. If the answer is an odd number, you get a point. And I'm even going to show you a diagram of the combinations . . ." Possible combinations for the products of 2 dice: 1 1 1 1 1 1 2 2 2 2 2 2 3x1 3x2 3x3 3x4 3x5 3x6 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 AIMS – statistics workshop Day 1 page 21 Whoa! . . . hold the phone! . . . the pesky Magician watched me diagram the possible outcomes. He saw my face when I discovered the game is NOT fair, so he is very surprised when I look him in the eye and say: "OK, Mr. Magician, I agree to play this game on one condition . . . " He is so startled to hear me say I will play that he says, "you will? I mean, oh, good. What is the condition?" "Every time there is an even number answer, you get one point. Every time there is an odd number answer, I get three points. This will adjust the game to make it fair. We will play until the first one scores 21 points. Agree?" AIMS – statistics workshop Day 1 page 22 To make an informed decision to play, 1. review the rules 2. picture the possibilities 3. regard the risks, and 4. state your chance of success and chances of losing. This will help you decide what you are willing to risk when playing a game. A game is FAIR only when it has equally likes outcomes of winning and loosing. Is Roulette FAIR? AIMS – statistics workshop Day 1 page 23 Multi-Stage Events & Tree Diagrams The probability of the outcome along any path is equal to the product of all probabilities along the path. Ex: Toss 2 coins 1st toss 2nd toss H H T H T T AIMS – statistics workshop Day 1 page 24 Multi-Stage Events & Tree Diagrams Ex: Roll 2 dice 1st die 1 2nd die 1 2 3 4 5 6 2 3 4 5 6 AIMS – statistics workshop Day 1 page 25 Multi-Stage Events & Tree Diagrams The probability of the outcome along any path is equal to the product of all probabilities along the path. Socks in the Dark: You have a drawer filled with single socks. If they were mated there would be 2 pair of white socks, 3 pair of tan socks, and 5 pair of black socks. Draw a tree diagram that shows all of the possible ways of picking two socks from the drawer in the dark one sock at a time, without replacement. Then determine the probability that you pick a pair of: a) black socks b) tan socks c) the same colored socks AIMS – statistics workshop Day 1 page 26 Tree Diagram for Socks-in-the-dark problem: 1st sock 2nd sock Black Black Tan White Tan Black Tan White White Black Tan White AIMS – statistics workshop Day 1 page 27 Most probability problems in the ISAT can be solved using tree diagrams, and without knowing rules of probability! Suppose a box contains 4 red and 5 white marbles. Two marbles are drawn. What is the probability they are both red? (note: you can draw any number of marbles; you can find any combination of colors; you can replace or not replace after each draw) 1st marble 2nd marble Red Red White Red White White AIMS – statistics workshop Day 1 page 28 Connected Math: Connections page 37 A bag contains several marbles. Some are red, some are white, and some are blue. Carlos counted the marbles and found that the theoretical probability of drawing a red marble is 1/6 and the theoretical probability of drawing a white marble is 1/3. 1. What is the smallest number of marbles that could be n the bag? 2. Could the bag contain 48 marbles? If so, how many of each color must it contain? 3. If the bag contains 4 red marbles and 8 white marbles, how many blue marbles must it contain? 4. How can you tell what the probability of drawing a blue marble is? AIMS – statistics workshop Day 1 page 29 Connected Math: Unit Reflections page 65 Joanna designed a game for the school carnival. She prepared two bags of marbles. Bag A contains 3 marbles: 1 red, 1 blue, 1 green. Bag B contains 4 marbles: 2 reds and 2 blues. To play the game, a contestant picks one marble form each bag. If the colors of the marbles match, the contestant wins a prize. These are the win/loss results for the first 30 games. W L L W W L L W W L L W W L L L W L L L L W L W L L L L W W a. What do these data suggest about the experimental probability of winning the game? b. What is the theoretical probability of winning the game? c. What explains the difference between your answers to question a and question b? AIMS – statistics workshop Day 1 page 30 A Probability Distribution is a list of the outcomes from an experiment along with their respective probabilities. Experiment: toss 2 coins Probability Distribution: Heads 0 1 2 P(H) 1/4 1/2 1/4 The only rules for a probability distribution is that each probability must be a number between 0 and 1, and the sum of all the probabilities must add to 1 (or 100%). AIMS – statistics workshop Day 1 page 31 Ex: Is the following a Probability Distribution? When four different households are surveyed on Monday night, the number of households with television tuned to Monday Night Football on ABC with their relative frequency is shown (based on data from Nielsen Media Research). MNF P(MNF) 0 0.522 1 0.368 2 0.098 3 0.011 4 0.001 AIMS – statistics workshop Day 1 page 32 Experiment: roll 2 dice Probability Distribution: sum P(sum) 1 2 3 4 5 6 7 8 9 10 11 12 0 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 AIMS – statistics workshop Day 1 page 33 13 0 Suppose a couple wants 4 children. Which is more likely, 3 of one sex and 1 of another, or 2 and 2 of each sex? Sample Space for 4 children (2x2x2x2 = 16 outcomes) GGGG BGGG What does the tree diagram look like? GGGB BGGB GGBG BGBG GGBB BGBB GBGG BBGG GBGB BBGB GBBG BBBG GBBB BBBB AIMS – statistics workshop Day 1 page 34 Make a probability distribution for the number of girls. Experiment: have 4 kids Probability Distribution: # girls Prob(# girls) 0 1 2 3 4 5 1/16 3/16 8/16 3/16 1/16 0 AIMS – statistics workshop Day 1 page 35 Probability distributions may result from discrete or continuous data. Tomorrow, we will look at a discrete probability distribution, the binomial distribution. Then we will look at a continuous probability distribution, the normal distribution. Probability Distributions Discrete * Binomial Geometric Continuous * Normal Chi-Square AIMS – statistics workshop Day 1 page 36 Poisson AIMS – statistics workshop Day 1 page 37