Download Unit #6 - Mattawan Consolidated School

Unit #6:
Counting, Probability & Inference
Name
Functions, Statistics, & Trigonometry: McKale
Goals & Objectives:
• S.IC.2 Decide if a specified model is consistent with results from a given -- data-generating process, e.g., using
simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a
row cause you to question the model?
• S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
• S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the
product of their probabilities, and use this characterization to determine if they are independent.
• S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B
as saying that the conditional probability of A given B is the same as the probability of A, and the conditional
probability of B given A is the same as the probability of B.
• S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and
everyday situations.
• S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret
the answer in terms of the model.
• S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
• S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B | A) =
P(B)P(A | B), and interpret the answer in terms of the model.
• S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
• S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
• S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling
a hockey goalie at the end of a game).
Enduring Understandings:
• The fundamental principles of probability are derived from basic principles of counting.
• Contingency tables can uncover relationships in data.
• Simulations can be used to estimate the probability of an event when the probability is difficult to calculate.
Essential Questions:
• How can basic laws of probability help us make sense of data?
• What are the basic counting strategies, and how are they used in calculating probabilities?
Purpose:
Probability can be represented numerically with a number between zero and one, with zero representing the
impossibility of something specific happening and one representing the certainty of that thing happening. This
number is often transferred to a percentage in order to predict something useful, such as a weather person
predicting the likelihood of rain on the following day. Probability differs from statistics because statistics
requires the examination of what has already occurred in order to determine how such a result occurred.
Without the use of statistics it would be very difficult to make decisions based on any collected data. Studying
statistics develops critical and analytic thinking skills allowing students to interpret data in a meaningful way.
However, the most important reason to study statistics is to be an informed consumer and voter. Like any other
tool, statistics can be used or misused. Yes, some individuals do actively lie and mislead with statistics. More
often though, many individuals unintentionally report erroneous statistical conclusions and an uniformed public
can accept their word as truth just because the numbers “say so.” If you know some of the basic statistical
concepts, you will be in a better position to evaluate the information that has been presented.
Lesson #6.1-6.2: (read section 6.1 – 6.2)
• S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events (“or,”
“and,” “not”).
• S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if they are
independent.
• S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in
terms of the model.
Warm-ups:
1. A drawer contains “r” red socks, “b” blue socks, and “w” white socks. Assume that you draw a
sock randomly from the drawer.
a) What is the total number of socks in the drawer?
b) What is the probability that the sock is red?
c) What is the probability that it is not white?
d) What is the probability that it is green?
2. According to Nielsen Media Research in October 2008, of the 113.1 million US households that
owned at least one TV set, 82% had 2 or more TV sets and 88% received basic cable. From this
information what are the largest and smallest percents of households that might have both 2 or
more TV sets AND also have basic cable?
Important Terms and Concepts: (define or describe the following terms or phrases and illustrate
their meaning)
TECHNICAL TERMINOLOGY & THEOREMS
IMPLICATIONS?
Experiment (p. 360)
Outcome (p. 360)
Sample Space (p. 360)
Event (p. 361)
Probability (p. 361)
Basic Properties of Probability (p. 363)
Odds (p. 363)
Mutually Exclusive (p. 367)
Union (p. 367)
Addition Counting Principle (p. 367)
Probability of the Union of Mutually Exclusive
Events (p. 368)
Intersection (p. 368)
General Addition Counting Principle (p. 369)
General Probability for Union of Events (p. 369)
Complementary Events (p. 370)
Probability of Complements (p. 371)
Lesson #6.1-6.2:
1. List the sample space for rolling a standard number cube. What is the probability of rolling a 4?
What is the probability of rolling a number greater than 2?
2. Consider the experiment of flipping two coins. What is the sample space for this event? What
is the probability of flipping exactly 2 heads? What are the odds of getting 2 heads?
3. Assume that the birth of boys and girls are equally likely. A family has 4 children, how many
outcomes are in the sample space? Find the probability that a family of four children has one
girl and three boys.
4. Two fair 8-sided dice with faces numbered 1-8, are rolled. Find each probability.
a. P( the sum is even)
b. P(the sum is odd but not 7)
c. P(the difference is less than 3)
d. P(the difference is a prime number)
5. Consider an experiment in which a coin with sides marked H and T is flipped, and a spinner
with 3 colors – red, green, and blue – is spun.
a. Give the experiment’s sample space.
b. List the outcomes in the event “the color spun is red.”
c. List the outcomes in the event “heads occur and the color spun is red.”
6. What is the probability of selecting a yellow or a blue marble from a box of 5 green, 3 yellow,
and 2 blue marbles?
7. What is the probability that a card drawn from a standard deck is either a face card or black?
8. Find the probability of rolling a 3 or a 6 on one toss of a die.
9. Find the probability of selecting a queen or a red card from a standard deck of cards.
10. Conrad tried out for both the volleyball team and the football team. The probability of his being
selected for the volleyball team is 4/5, while the probability of his being selected for the football
team is ¾. The probability of his being selected for both teams is 7/10. What is the probability
that Conrad will be selected for either the volleyball team or the football team?
11. Jameel and Rhonda had to take a make up test after school, their teacher told them they could
come in no earlier than 3:00 and leave no later than 4:00. Jameel took 38 minutes on the test
and Rhonda 51 minutes. For at least how many minutes were Jameel and Rhonda taking the test
at the same time?
12. A pair of six-sided dice is thrown. If the dice are fair, …
a) What is the probability that the dice show
a sum of 7 or 11?
b) What is the probability that the dice show
doubles (two numbers alike) or a sum of 8?
Problem Set
#6.1-6.2
Description
6.1 #1-14
6.2 #1-6, 9, 10, 12-15, 17-18
Due Date
Questions?
Lesson #6.3-6.4, 10.1: (read section 6.3-6.4 & 10.1)
• S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) =
P(A)P(B | A) = P(B)P(A | B), and interpret the answer in terms of the model.
• S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and
solve problems.
Warm-ups:
1. Write down all the three-letter combinations that begin with D using the letters A, B, C, D, and
E only once.
2. How many three-letter combinations are there using the letters A, B, C, D, and E only once?
3. Ten permutations (arrangements) of the letters AELST are legal words in the game of Scrabble.
Name as many as you can.
4. List all permutations of 3 letters taken from the 4 letters C, O, M, and B.
5. Group the permutations of problem #4 so that those permutations that use the same letters (such
as BCM and CMB) are in the same group.
a) How many groups are there?
b) How many are in each group?
Important Terms and Concepts: (define or describe the following terms or phrases and illustrate
their meaning)
TECHNICAL TERMINOLOGY & THEOREMS
IMPLICATIONS?
Multiplication Counting Principle (p. 374)
String (p. 375)
Length (p. 375)
Strings with Replacement Theorem (p. 376)
I
Independent Events (p. 377)
Calculating Permutations (p. 382)
Definition (p. 384)
Combination (p. 618)
Calculating Combinations (p. 619)
Lesson #6.3-6.4:
1. How many 7-digit telephone numbers can be formed if the first digit cannot be 0 or 1?
2. How many ways can the 4 letters of a radio station be arranged if the first letter must be W or K
and no letters can be repeated?
3. A golf manufacturer makes irons with 7 different shaft lengths, 3 different grips, and 2 different
club head materials. How many different combinations are offered?
4. A lottery game played in some states involves picking 3 digits from 0 to 9 in order. Describe a
sample space for this experiment and determine the number of elements in the sample space.
5. Three of the questions on a science test are multiple choice with four choices each.
a. How many ways are there to answer these questions?
b. If you guess randomly on each of these questions, what is the probability of answering all
these correctly?
6. Some states allow license plates with 3 letters followed by 4 digits from 0 through 9. How many
license plates are possible?
7. In a certain spinning wheel, there are 20 sectors of equal size. In 18 of these sectors you win a
prize, but in the other 2, you lose all your winnings. If the wheel spins randomly, what is the
probability of winning five prizes and then losing on the sixth spin?
8. Using a standard deck of playing cards, find the probability of drawing a king, replacing it, and
then drawing a second king.
9. In volleyball, a team plays 6 players at a time, three at the net and three behind. How many
different ways are there to arrange three at the net from the six who are playing?
10. In a class of 24 students, three are selected to serve on a teacher advisory group. How many
advisory groups can be created?
11. An art gallery has 12 paintings by Renaissance painters but only enough room to show 8 of
them. In how many ways can they arrange 8 of the 12 paintings in the 8 places they have for
them?
12. How many six-letter strings of letters can be formed from six letters in the word
PALINDROME without replacement?
13. An ice cream stand has 5 flavors of ice cream. If you wish to get a 3 scoop bowl special and
you must get 3 different flavors, how many ways can you order your ice cream bowl?
14. A box contains 12 black and 8 green marbles. How many ways can 3 black and 2 green marbles
be chosen?
15. A restaurant menu has ten appetizers and twelve main courses. A group of people decides they
want to order six appetizers and four main courses. In how many ways can they do this?
16. To win a particular lottery game, it is necessary to match the numbers on 5 balls that are
randomly picked from 50 balls numbered 1 to 50 and the number on a special black ball that is
picked from 9 other balls numbered 1 to 9. What is the probability that a single lottery ticket
will match the 5 balls and the special black ball?
17. A restaurant menu has ten appetizers and twelve main courses. A group of people decides they
want to order six appetizers and four main courses. In how many ways can they do this?
Problem Set
Description
#6.3-6.4, 10.1
6.3 #1-3, 6-10
6.4 #1-2, 4-8, 13, 14ab, 15
10.1 #1-6c, 9, 10, 14, 15, 19, 20
Due Date
Questions?
Lesson #6.5-6.6: (read sections 6.5-6.6)
• S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.
• S.CP.5 Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
• S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also
belong to A, and interpret the answer in terms of the model.
Warm-ups:
1. Willie Fielder hurt himself two games into the 2047 baseball season and only batted 6 times
with 1 hit, for a batting average of .167. Scott Scrub played the entire season but was a second
stringer, so was up only 100 times and got 19 hits, for a batting average of .190. In the 2048
season, Willie was well and got 201 hits in 600 at bats. Scott still remained a second stringer
and was up only 100 times again, but got 35 hits.
a) What was Willie's batting average for the 2048 season?
b) What was Scott's batting average for the 2048 season?
c) What was Willie's combined batting average for both seasons?
d) What was Scott's combined batting average for both seasons?
2. Suppose 60% of the singers in a school play are in the school choir. In the school as a whole,
suppose 10% of the students are in the choir and 5% are in the school play. Finally, suppose
there are 600 students in the school.
a) If a student in the play is randomly chosen, what is the probability that student is in the choir?
b) If a student in the choir is randomly chosen, what is the probability that student is in the
play?
c) If a student in the school is randomly chosen, what is the probability that student is in both
the play and the choir?
Important Terms and Concepts: (define or describe the following terms or phrases and illustrate
their meaning)
TECHNICAL TERMINOLOGY & THEOREMS
IMPLICATIONS?
Contingency Table (p. 387)
Conditional Probability (p. 395)
Lesson #6.5-6.6:
1. Use Titanic Table 1 shown below. Round each answer to the nearest percent.
a. Out of all the people on the ship,
what percent died?
b. What percent of passengers in third
class died?
c. What percent of passengers in first or second class died?
2. Use the data shown to the right. Round each answer to the nearest tenth of a percent.
a. What percent of the people in the study did not have a tattoo?
b. What percent of people in the study with no tattoo had Hepatitis C?
c. What percent of people in the study with a tattoo had Hepatitis C?
3. Let B = a person eats a good breakfast; let L = a person eats a good lunch. Suppose, in a group
of 80 people, 43 eat good breakfasts and good lunches, 21 eat a good breakfast but not a good
lunch, 12 eat a good lunch but not a good breakfast, and the rest eat neither a good lunch nor a
good breakfast.
a. Find P(eat good lunch and good breakfast)
b. Find the probability that a person ate a good lunch given that they ate a good breakfast.
c. Find the probability that a person ate a good breakfast given that they ate a good lunch
4. Each of four boxes contains a red marble and a yellow marble. A marble is selected from each
box without looking. What is the probability that exactly three red marbles are selected if the
third marble is known to be red?
5. Two dice are tossed. Find the probability that the numbers showing on the cubes match, given
that their sum is greater than 7.
6. Three coins are tossed. Find the probability that exactly two coins show tails, given that the
third coin shows tails.
7. A card is chosen from a standard deck of cards. Find each probability, given that the card is red.
a) P(diamond)
b) P(six of hearts)
c) P(queen or 10)
d) P(face card)
8. A survey taken at Stirers High School shows that 48% of the respondents like soccer, 66% like
basketball, and 38% like hockey. Also, 30% like soccer and basketball, 22% like basketball and
hockey, and 28% like soccer and hockey. Finally, 12% like all three sports.
a) If Meg likes basketball, what is the probability that she also likes soccer?
b) If Jaime likes soccer, what is the probability that he also likes hockey and basketball?
c) If Ashley likes basketball, what is the probability that she also likes hockey?
d) If Brett likes soccer, what is the probability that he also likes basketball?
Problem Set
#6.5-6.6
Description
6.5 #1, 2, 7-10
6.6 #worksheet 6-6
Due Date
Questions?
Lesson #6.7-6.8: (read section 6.7-6.8)
• S.IC.2 Decide if a specified model is consistent with results from a given -- data-generating
process, e.g., using simulation. For example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to question the model?
• S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
• S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Warm-ups:
1. Toss a coin until you get 3 heads and 3 tails. How many tosses did it take? Compare your
results with others in your class. What do you think is the number of tosses that it takes on
average to get 3 heads and 3 tails.
2. You toss a coin 8 times and get 8 heads.
a) Explain why getting a head or a tail next time might be equally likely.
b) Explain why heads might be more likely than tails the next time you toss.
c) Explain why someone might think that tails is more likely than heads the next time you toss.
3. In a certain metropolitan area, 250,000 people live in the city and 400,000 live in the suburbs.
When the basketball playoffs started, there are 32 teams. Of those teams, 14 were from the city
and 18 were from the suburbs. How many would have been “expected” to be from the city and
how many from the suburbs if the numbers of teams were proportional to the population?
Important Terms and Concepts: (define or describe the following terms or phrases and illustrate
their meaning)
TECHNICAL TERMINOLOGY & THEOREMS
IMPLICATIONS?
Simulation (p. 402)
Trial (p. 402)
Monte Carlo Method (p 403)
Expected Count (p. 410)
Law of Large Numbers (p. 411)
Lesson #6.7-6.8:
1. Suppose a town has about 25,000 adults in its population. If the unemployment rate in the town
is 5%, and you sampled 200 adults from the town at random, what is your expected number of
unemployed adults?
2. Read page 402 and the top of page 403, then complete Activity 1 (Steps 1 through 3) on page
403.
Step 1: Perform one trial of the simulation. How many boxes did it take you to get all four
Sports Stars cards? How does your result compare with those of your classmates?
Step 2: Bobby's parents refuse to buy dozens of boxes. Based on your results for Step 1, pick a
number of boxes B that you think is likely to give Bobby a full set of Sports Stars cards and
also satisfy his parents.
Step 3: Simulate buying B boxes ten times, noting each time whether you were successful in
finding all four cards. Record your data in a table.
Step 4: Combine your data with the rest of the class. Construct a graph showing the percent of
times that buying x boxes resulted in having all 4 cards.
Step 5: Do you think it would be worth it to buy lots of boxes to attempt to obtain all four
cards? Why or why not?
3. Read page 404 and Example 1. Now suppose that, in the past, 95% of people who reserve a
seat on a particular flight actually show up. If the plane holds 118 passengers, describe how
you could use a simulation to estimate how many passengers will show up?
4. Activity 2: Two brown-eyed parents can have a blue-eyed child if they each carry a recessive
gene for blue eyes. Assume that in these cases, the probability of blue eyes is ¼ and the
probability of brown eyes is ¾. Design a simulation to estimate the probability of having
exactly one of three children with brown eyes.
Step 1: Assign the digits 1, 2, 3, and 4 to represent the eye color when one child is born.
Step 2: Simulate forty trials of three-child families and record the results.
Step 3: Use the corresponding relative frequency to estimate the probability that a family with
three children has exactly one with brown eyes.
5. A college has places for 280 entering freshman. Typically, 30% of freshman students offered
admission choose not to enroll. To compensate for this the dean of admissions offers admission
to 385 students. Design a simulation to estimate the probability that more than 280 will enroll.
6. A current estimate for the proportion of left-handers in the population is 9%. What is the
expected count of left-handers in a classroom of 36 students?
7. During WWII, John Kerrich, a South African mathematician, was captured by the Germans and
interned at a camp. To pass the time, he did experiments on chance processes. In one such
experiment he tossed a coin 10,000 times and observed 5067 heads.
a) What was the expected number of heads?
b) Find the difference between the observed and expected numbers of heads.
c) What was the relative frequency of heads?
d) What was the difference between the relative frequency and the probability of heads?
Problem Set
#6.7-6.8
Description
6.7
#3ab, 5, 6, 9, 10
6.8 #1-12
Due Date
Questions?