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Math 2
Statistics
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Module 9: Probability CCS Standard
1 Basic Probability S-CP 1
2 "Or (+)" Probability Exclusive/Inclusive: "And(*)" Probability Independent S-CP 2, 7
3 "Or (+)" Probability Exclusive/Inclusive: "And (*)" Probability Independent S-CP 2, 7
4 MVP 9.1 S-CP 1,2,7
5 MVP 9.2 S-CP 6
6 MVP 9.3 S-CP 6
7 Review
8 Mid Unit Test
9 MVP 9.4 S-CP 2, 6
10 MVP 9.5 S-CP 6,7
11 MVP 9.6 S-CP 2-5
12 Factorial, Combination, Permutation S-CP 9
13 Factorial, Combination, Permutation S-CP 9
14 Probability: Factorial, Combination, Permutation S-CP 9
15 Probability: Factorial, Combination, Permutation S-CP 9
16 Test
17 Performance Task
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_
Lesson 1: Definitions for ProbabilityProbability
Probability is the likelihood of the occurrence of an event. The probability of event A is
written P(A). Probabilities are always numbers between 0 and 1, inclusive.
The four basic rules of probability:
1) For any event A, 0 ≤ P(A) ≤ 1.
2) P(impossible event) = 0.
3) P(sure event) = 1.
4) P(not A) = 1 - P(A).
Also written P(complement of A) = 1-P(A) or
Pc(A) = 1 - P(A) or
P (A) = 1 - Pc (A)
Experiment or Trial:
An action where the result is uncertain.
Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.
Sample Space
A set of all possible outcomes of an experiment For a coin toss: S = {Head, Tail} For Dice roll: S = {1, 2, 3, 4, 5, 6} For two coins: S = {HH, HT, TH, TT}
Outcome
A single specific possible result of an experiment.
Event
A result from a particular experiment.
Outcomes are what could happen, events are what did happen and can combine multiple outcomes (tossing a coin and rolling a die).
Experiment
Tossing a coin
Rolling a six sided
Outcome Example
Heads
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The probability of an event is the chance that it will occur, expressed as a ratio
of a specific event to all possible events.
Or....
Probability = number of actual events
number of possible events
Or....
Probability = number of favorable outcomes
number of possible outcomes
The probability of an event that is certain to occur is 1.
The probability of an event that is impossible to occur is 0.
The probability of an event that is possible is >0 and <1
Problem: A spinner has 4 equal sectors colored yellow, blue, green and red. What are
the chances of landing on blue after spinning the spinner? What are the
chances of landing on red?
Solution: The chances of landing on blue are 1 in 4, or one fourth.
The chances of landing on red are 1 in 4, or one fourth.
In order to measure probabilities, mathematicians have devised the following formula for finding the
probability of an event.
The probability of event A is the number of ways event A can occur divided by the total number of
possible outcomes. Let's take a look at a slight modification of the problem from the top of the page.
Experiment 1: A spinner has 4 equal sectors colored yellow, blue, green and red.
After spinning the spinner, what is the probability of landing on each
color?
Outcomes: The possible outcomes of this experiment are yellow, blue, green,
and red.
Probabilities: P(yellow) =
# of ways to land on yellow =
1
total # of colors 4
P(blue) = # of ways to land on blue
= 1
total # of colors 4
P(green) = # of ways to land on green
= 1
total # of colors 4
P(red) = # of ways to land on red
= 1
total # of colors 4
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Experiment 2: A single 6-sided die is rolled. What is the
probability of each outcome? What is the
probability of rolling an even number? of rolling
an odd number?
Outcomes: The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6.
Probabilities: P(1) =
# of ways to roll a 1 =
1
total # of sides 6
P(2) = # of ways to roll a 2
= 1
total # of sides 6
P(3) = # of ways to roll a 3
= 1
total # of sides 6
P(4) = # of ways to roll a 4
= 1
total # of sides 6
P(5) = # of ways to roll a 5
= 1
total # of sides 6
P(6) = # of ways to roll a 6
= 1
total # of sides 6
P(even) = # ways to roll an even number
= 3 =
1
total # of sides 6 2
P(odd) = # ways to roll an odd number
= 3 =
1
total # of sides 6 2
Experiment 2 illustrates the difference between an outcome and an event. A single outcome of this
experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. Rolling an even number (2, 4 or 6) is an event,
and rolling an odd number (1, 3 or 5) is also an event.
In Experiment 1 the probability of each outcome is always the same. The probability of landing on each
color of the spinner is always one fourth. In Experiment 2, the probability of rolling each number on the
die is always one sixth. In both of these experiments, the outcomes are equally likely to occur. Let's look
at an experiment in which the outcomes are not equally likely.
Experiment 3: A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If
a single marble is chosen at random from the jar, what is the
probability of choosing a red marble? a green marble? a blue
marble? a yellow marble?
Outcomes: The possible outcomes of this experiment are red, green, blue and
yellow.
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Probabilities: P(red) =
# of ways to choose red =
6 =
3
total # of marbles 22 11
P(green) = # of ways to choose green
= 5
total # of marbles 22
P(blue) = # of ways to choose blue
= 8
= 4
total # of marbles 22 11
P(yellow) = # of ways to choose yellow
= 3
total # of marbles 22
The outcomes in this experiment are not equally likely to occur. You are more likely to
choose a blue marble than any other color. You are least likely to choose a yellow marble.
Experiment 4: Choose a number at random from 1 to 5. What is the probability of each outcome?
What is the probability that the number chosen is even? What is the probability that the
number chosen is odd?
Outcomes: The possible outcomes of this experiment are 1, 2, 3, 4 and 5.
Probabilities: P(1) =
# of ways to choose a 1 =
1
total # of numbers 5
P(2) = # of ways to choose a 2
= 1
total # of numbers 5
P(3) = # of ways to choose a 3
= 1
total # of numbers 5
P(4) = # of ways to choose a 4
= 1
total # of numbers 5
P(5) = # of ways to choose a 5
= 1
total # of numbers 5
P(even) = # of ways to choose an even number
= 2
total # of numbers 5
P(odd) = # of ways to choose an odd number
= 3
total # of numbers 5
The outcomes 1, 2, 3, 4 and 5 are equally likely to occur as a result
of this experiment. However, the events even and odd are not
equally likely to occur, since there are 3 odd numbers and only 2
even numbers from 1 to 5.
Summary: The probability of an event is the measure of the chance that the event will occur as a result
of an experiment. The probability of an event A is the number of ways event A can occur
divided by the total number of possible outcomes. The probability of an event A, symbolized
by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event.
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Describing Events
The 10 cards below provide information on the ten students on the robotics team at a high school. For each student, the
following information is given:
The students are identified by an ID number: S1, S2, …, S10.
Gender
Grade level
Whether or not the student is currently enrolled in a science class
Whether or not the student is currently participating on a school sports team
Typical number of hours of sleep on a school night
It might be helpful to cut out these cards so that you can sort and rearrange them as you answer the following
questions.
a. Suppose that one student on the robotics team will be selected at random to represent the team at an upcoming
competition. This can be viewed as a chance experiment. Which one of the following is the sample space for this
experiment?
S = {Student ID, gender, grade level, science class, sports team, hours of sleep}
S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10}
S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10, male, female, 9, 10, 11, 12, yes, no, 6, 7, 8, 9}
b. What outcomes from the sample space are in the event that the selected student is taking a science class?
Consider the following three events:
A = the selected student is female
B = the selected students is on a sports team
C = the selected student typically sleeps less than 8 hours on a school night
d. For each of these three events, list the outcomes that make up the event.
e. From the three events above, which outcomes are in the following events?
i. A or B
ii. A and C
ii. not C
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f. Describe each of the events in question 4 in words.
g. Define two additional events in the context of this chance experiment. Use the letters D and E to represent these
events.
h. For the two events you defined in question 6, describe the event D or E and the event D and E in words.
i. Is the set of outcomes in D or E and the set of outcomes in D and E the same for the two events you have defined?
Explain why or why not.
Solution
a. The correct choice is S = {S1, S2, S3, S4, S5, S6, S7, S8, S9, S10 }. Because the chance experiment consists of
selecting one of the 10 students at random, the sample space would consist of 10 outcomes corresponding to the 10
students in the robotics club.
b. S1, S4, S5, S6, S7, S10
c. A: S1, S5, S6, S8
B: S3, S4, S5, S9
C: S1, S4, S5, S7, S9
d. A or B: S1, S3, S4, S5, S6, S8, S9
A and C: S1, S5
not C: S2, S3, S6, S8, S10
e. A or B: the event that the selected student is female or is on a sports team
A and C: the event that the selected student is female and typically sleeps less than 8 hours on a school night not C: The
event that the selected student does not typically sleep less than 8 hours on a school night. The event not C could also be
described as the event that the selected student typically sleeps at least 8 hours on a school night.
The answers for questions 6 – 8 will vary depending on what events that the student defines. Check to make sure that
students have given a correct description of the union and the intersection for the two events that they defined. In question 8,
it might be possible for students to define two events for which both the union and the intersection are the same, but this
would not usually be the case. Check the explanation carefully if a student answers yes for question 8.
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S1
Gender: Female
Grade: 11
Science course: yes
Sports team: no
Sleep: 7 hours
S2
Gender: Male
Grade: 9
Science course: no
Sports team: no
Sleep: 9 hours
S3
S4
Gender: Male Gender: Male
Grade: 11 Grade: 10
Science course: no Science course: yes
Sports team: yes Sports team: yes
Sleep: 8 hours Sleep: 6 hours
S5
Gender: Female
Grade: 12
Science course: yes
Sports team: yes
Sleep: 7 hours
S6
Gender: Female
Grade: 9
Science course: yes
Sports team: no
Sleep: 9 hours
S7
S8
Gender: Male Gender: Female
Grade: 11 Grade: 10
Science course: yes Science course: no
Sports team: no Sports team: no
Sleep: 7 hours Sleep: 8 hours
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HOMEWORK
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Name: _____________________________________________________________Date _______Pd:___ Topic: Probability Word Problems- What is the probability? Homework For the following problems
Write the sample space of the experiment
List the favorable outcomes of the experiment
Calculate the probability
1. Jill is playing cards with her friend when she draws a card from a pack of 20 cards numbered from 1 to 20. What
is the probability of drawing a number that is square?
2. Each of the letters in the word SAMSUNG are on separate cards, face down on the table. If you pick a card at
random, what is the probability that its letter will be S or U?
3. You roll a SIX sided die. What is the probability that the value of the roll will be five?
4. A bag contains 5 blue sticks, 4 red sticks, and 3 orange sticks and you ask a friend to pick one without looking.
What is the probability that the stick will be blue?
5. You think of a number from the first twenty negative integers.
6. What is the probability that the integer chosen will be divisible by 4?
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7. Each of letters in the word OPPORTUNITIES are on separate cards, face down on the table. If you pick a card at
random, what is the probability that its letter will be O or I?
8. A bag contains 6 purple sticks, 2 orange sticks, and 4 black sticks and you ask a friend to pick one without
looking. What is the probability that the stick will be purple?
9. A bag contains 3 green marbles, 5 blue marbles, and 8 red marbles and you ask a friend to pick one without
looking. What is the probability that the marbles will be orange?
10. A bag contains 5 green shirts, 9 blue shirts, and 10 red shirts and you ask a friend to pick one without looking.
What is the probability that a shirt will be picked?
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Lesson 2: Compound Probability
Compound Events That Are
When two events cannot happen at the same time, they are mutually exclusive events.
For example, you have a die and you are asked to find the probability of rolling a 1 or a 2. You know when you roll the die, only one of those numbers can appear, not both. Therefore, these events are mutually exclusive of each other.
Mutually Exclusive Events (Events that cannot happen at the same time) P(A or B) = P(A) + P(B)
Take note: With this formula, you are adding the probabilities of each event, not multiplying. (“or” means add!!!)
Let's take a look at an example of mutually exclusive events.
Example 1
Mutually exclusive events are pretty straightforward. Now let's take a look at compound events that are inclusive.
When two events can occur at the same time, they are .
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For example, let's take our example of rolling a regular 6-sided die. You are asked to find the probability of rolling a 2 or an even number. These events are inclusive because they can happen at the same time. A 2 is an even number, so this would satisfy both, but you could also roll a 4 or 6. Because 2 is an even number, these are inclusive events.
Inclusive Events (Events that CAN happen at the same time)
P(A or B) = P(A) + P(B) - P(A and B)
This is a little trickier, so let's take a look at example of inclusive events.
Example 2
I know that compound events can be confusing, but first you must determine if the events are exclusive or inclusive.
If the events are exclusive, then just add the probabilities of each individual event.
If the events are inclusive, you must remember to subtract the number outcomes that occur in both events.
- See more at: http://www.algebra-class.com/probability-lessons.html#sthash.0pLWQZbG.dpuf
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http://www.regentsprep.org/regents/math/algebra/APR6/Lindep.htm
http://www.algebra-class.com/probability-problems.html#sthash.a22Who2Z.dpuf
Example 1
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Example 2
- See more at:
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Name:________________________________________Date______________Pd._______ Topic: Single and Compound Events – Classwork/Homework Complete the following problems 1. 3 coins are tossed and two six-sided dice are rolled. What is the probability of getting 3 heads and an even number on a dice? 2. A bag contains 13 cars: 2 are blue cars and 11 are red cars. What is the probability for picking a blue car? 3. A dice is rolled 6 times. What is the total number of possible outcomes? 4. A Box contains 20 necklaces: 12 are Gold and 6 are Silver. What is the probability for picking a Silver necklace? 5. A bag contains ten black marbles, twenty white marbles, and five grey marbles. You pick one without looking. What is the probability that the marble will be either white OR black? 6. A bag contains fourteen yellow balls, two violet balls, and nine red balls. You pick one without looking. What is the probability that the marble will be either yellow OR violet? 7. A bag contains thirteen blue sticks, five pink sticks, and nine red sticks. You pick one without looking. What is the probability that the stick will be either blue OR red? 8. Eva forgot to study one of the chapters for her history test so she had to guess on two multiple-choice questions which each had four answer choices. What is the probability that she got both questions correct? 9. Rennin has a pair of dice and he throws them on the table then, what is the probability that the sum of the numbers appears is 6? 10. Three fair coins are tossed what is the probability that tails will come up for all of them?
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11. A pair of dice is rolled. What is the probability that the sum of the numbers rolled is 7? 13. A box contains 2 red marbles, 3 white marbles, 4 green marbles and 1 blue marble. Two marbles are drawn at random without replacement. Find the probability of
a. Selecting a green marble on the second draw if the first marble is blue.
b. Selecting a white marble on the first draw and red marble on the second draw.
c. Selecting a red marbles on both draws.
d. Selecting a red or white on the first draw and green or blue on the second draw.
e. Selecting a white marble on the first draw and a white or blue on the second draw.
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Name:________________________________________Date______________Pd._______ Classwork/Homework
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Extra review
Gender of Children
Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl.
Use a tree diagram to find the sample space for the gender of three children in a family
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Tuberculosis (TB) can be tested in a variety of ways, including a skin test. If a person has
tuberculosis antibodies, then they are considered to have TB. Below is a tree diagram
representing data based on 1,000 people who have been given a skin test for tuberculosis.
1. Use your knowledge to write several probability statements about this test (based on
the numbers provided).
2. Look over the statements you wrote. Put an asterisk (*) next to those that are
conditional probability statements (statements based on margin “row” or “column”
percentages). If there are not any, add some now.
3. Part of understanding the world around us is being able to take information, make sense
of it, and then explain it to others. Based on your statements above, what would you say
to a friend regarding the validity of their results if they are testing for TB and only get a
skin test? Be sure to use data to best inform your friend.
9.1 TB or Not TB? A Develop Understanding Task
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Other questions to consider….
4. In this situation, explain the consequences of errors (having a test with incorrect results).
5. If a health test is not 100% certain, why might it be beneficial to have the results lean more
toward a false positive?
6. Is a sample space of 200 enough to indicate whether or not this is true for an entire
population?
7. How would you answer the young adult who tested positive and asks, “Do I really have TB?”
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For each Venn Diagram provided answer the questions.
1. How many students were surveyed?
2. What were the students asked?
3. How many students are in
both choir and band?
4. How many students are not in
either choir or band?
5. What is the probability that a
randomly selected student would be in band?
This Venn Diagram represents enrollment in some of the elective courses.
6. What does the 95 in the center tell you?
7. What does the 145 tell you?
8. How many total students are represented in the diagram?
9. Which elective class has the least number
of students enrolled?
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Set Topic: Interpret a tree diagram, making observations of probability.
Given the tree diagram below answer the questions and determine the probabilities. The diagram represents the number of plate appearances during the first month of a minor league baseball season.
10. How many times did a batter come to the plate during
this time period?
11. Based on this data, if you are a left4handed batter what is
the probability that you will face a right4handed pitcher?
12. Based on this data, if you are a right4handed batter what
is the probability that you will face a left4handed pitcher?
13. What is the probability that a left4handed pitcher will be throwing for any given plate
appearance?
14. What is the probability that a left4handed batter would be at the plate for any given plate
appearance?
15. What observations do you make about the data? Is there any amount that seems to be
overly abundant? What might account for this?
Go Topic: Basic Probability
Find the probability of achieving success with each of the events below.
16. Rolling an even number on standard six4sided die.
17. Drawing a black card from a standard deck of cards.
18. Flipping a coin and getting Heads three times in a row.
19. Rolling a die and getting a four.
20. Drawing an ace from a deck of cards.
21. Rolling a die twice in a row and getting two threes.
22. From a bag containing 3 blue, 2 red, and 5 white marbles. Pulling out a red marble.
obabili 9.
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9.2 Chocolate versus Vanilla A Solidify Understanding Task
Danielle loves chocolate ice cream much more than vanilla and was explaining to
her best friend Raquel that so does most of the world. Raquel disagreed and
thought vanilla is much better. To settle the argument, they created an online
survey asking people to choose their favorite ice cream flavor between chocolate
and vanilla. After completing the survey, the following results came back:
There were 8,756 females and 6,010 males who responded. Out of all the males, 59.7% chose vanilla over chocolate. 4,732 females chose chocolate.
1. Upon first observations, which flavor do you think “won”?
Write a sentence describing what you see at ‘first glance’ that makes you think this.
2. Raquel started to organize the data in the following two-way table. See if you can help complete this (using counts and not percentages):
Chocolate Vanilla Total
Female 8,756
Male 6,010
Total
3. Organize the same data in a Venn diagram and a tree diagram.
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4. Using your organized data representations, write probabilities that help support your claim regarding the preferred flavor of ice cream. For each probability, write a complete statement as well as the corresponding probability notation.
5. Looking over the three representations (tree diagram, two-way table, and Venn diagram), what probabilities seem to be easier to see in each? What probabilities are hidden or hard to see?
Highlighted (easier to see) Hidden Tree diagram Tree diagram
Two-way table Two-way table
Venn diagram Venn diagram
6. Getting back to ice cream. Do you think this is enough information to proclaim the statement that one ice cream is favored over another? Explain.
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Topic: Analyzing data in a Venn Diagram Use the Venn Diagrams below to answer the following questions. (Hint: you may use the same data provided in the two-way table from question 3 on the next page to help make sense of the Venn Diagram)
The following Venn Diagram represents the relationship between favorite sport (Soccer or Baseball) and gender (Female or Male).
1. How many people said soccer is their favorite sport?
2. How many females are in the data?
3. How many males chose baseball?
4. What is the probability that a person would say soccer is their
favorite sport? P(soccer) =
5. What is the probability that a female would say soccer is their favorite sport? (“Out of all females,
% say soccer is their favorite sport”) P(soccer | female) = The following Venn Diagram represents the relationship between favorite subject (Math or Science) and grade level (Ninth or Tenth). Using this data, answer the following questions.
6. How many people said math is their favorite subject?
7. How many tenth graders are in the data?
8. How many ninth graders chose science?
9. What is the probability that a person would say science is their favorite subject? P(s) = 30
10. What is the probability that a tenth grader would say science is their favorite subject? (“If you
are a tenth grader, then the probability of science being your favorite subject is %”) P(science | tenth)=
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30
Probability 9.2
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Set Topic: Writing conditional statements from a two-way table.
11. Complete the table and write three conditional statements.
Soccer Baseball Total Male 30
Female 50 76 Total 85
12. Complete the table about preferred genre of reading and write three conditional statements.
Fiction
Non4 Fiction
Total
Male 10
Female 50 60 Total 85
13. Complete the table about favorite color of M&M’s and writes three conditional statements.
Blue Green Red Other Total Male 15 20 15 60 Female 30 20 10
Total 45 130
14. Use the information provided to make a tree diagram, a two4way table and a Venn Diagram.
Data was collected at the movie theater last fall. Not about movies but clothes.
6,525 people were observed.
3,123 had on shorts and the rest had on pants
45% of those wearing shorts were denim.
Of those wearing pants 88% were denim.
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Go Topic: Fractions, Percent and there operations
Find the desired values.
15. What is half of one-third? 16. What is one4third of two-fifths?
17. What is one-fourth of four-sevenths? 18. What percent is 5/8 ?
19. What is 35% of 50? 20. Seventy is 60% of what number?
21. Write 7/12 as a percent. 22. Write 1/6 as a percent.
23. What is 52% of 1,200? 24. What percent is 32 of 160?
25. Sixty is what percent of 250? 26. What percent of 350 is 50?
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9.3 Fried Freddy’s A Solidify Understanding Task
Danielle was surprised by the results of the survey to
determine the ‘favorite ice cream’ between chocolate and
vanilla (See task 9.2 Chocolate vs. Vanilla). The reason, she
explains, is that she had asked several of her friends and the
results were as follows:
Chocolate Vanilla Total
Female 23 10 33
Male 6 8 14
Total 29 18 47
1. In this situation, chocolate is most preferred. How would you explain to her that this
data may be less ‘valid’ compared to the data from the online survey?
Using a sufficiently large number of trials helps us estimate the probability of an event happening. If
the sample is large enough, we can say that we have an estimated probability outcome for the
probability of an event happening. If the sample is not randomly selected (only asking your friends)
or not large enough (collecting four data points is not enough information to estimate long run
probabilities), then one should not estimate large scale probabilities. Sometimes, our sample
increases in size over time. Below is an example of data that is collected over time, so the estimated
probability outcome becomes more precise as the sample increases over time.
Freddy loves fried food. His passion for the perfect fried food recipes led to him opening the
restaurant, “Fried Freddy’s.” His two main dishes are focused around fish or chicken. Knowing he
also had to open up his menu to people who prefer to have their food grilled instead of fried, he
created the following menu board:
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After being open for six months, Freddy realized he was having more food waste than he should because he
was not predicting how much of each he should prepare in advance. His business friend, Tyrell, said he could
help.
2. What information do you think Tyrell would need?
Luckily, Freddy uses a computer to take orders each day so Tyrell had lots of data to pull from. After
determining the average number of customers Freddy serves each day, Tyrell created the following Venn
diagram to show Freddy the food preference of his customers:
To make sense of the diagram, Freddy computed the following probability statements:
3. What is the probability that a randomly selected customer would order fish?
P(fish) =
Shade the part of the diagram that models this solution.
4. What is the probability that a randomly selected customer would order fried
fish?
P(fried ∩ fish) = P(fried and fish) =
Shade the part of the diagram that models this solution.
5. What is the probability that a person prefers fried chicken?
P(fried ∩ chicken) = P(fried and chicken) =
Shade the part of the diagram that models this solution.
Fried Fish
20% 15% 30%
Grilled 35%
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6. What is the estimated probability that a randomly selected customer would want their
fish grilled?
P(grilled and fish) = P( ) =
Shade the part of the diagram that models this solution.
7. If Freddy serves 100 meals at lunch on a particular day, how many orders of fish should he
prepare with his famous fried recipe?
8. What is the probability that a randomly selected person would choose fish or fried?
P(fried ⋃ fish) = P(fried or fish) =
Shade the part of the diagram that models this solution.
9. What is the probability that a randomly selected person would NOT choose fish or
fried?
Shade the part of the diagram that models this solution.
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Ready Topic: Independent and Dependent events
In some of the situations described below the first event effects the subsequent event (dependent events). In others each of the events is completely independent of the others (independent events). Determine which situations are dependent and which are independent.
1. A coin is flipped twice. The first event is the first flip and the second event is the next flip.
2. A bag of marbles contains 3 blue marbles, 6 red marbles and 2 yellow marbles. Two of the marbles are drawn out of the bag. The first even is the first marble taken out the second event is the second marble taken out.
3. An attempt to find the probability of there being a right-handed or a left-handed batter at the plate
in a baseball game. The first event is the 1st batter to come to the plate. The second event is the second player to come up to the plate.
4. A standard die is rolled twice. The first event is the first roll and the second event is the second
roll.
5. Two cards are drawn from a standard deck of cards. The first event is the first card that is
drawn the second event is the second card that is drawn.
Set Topic: Addition Rule, interpreting a Venn Diagram
6. Sally was assigned to create a Venn diagram to represent P(A ⋃ B). Sally first writes
P(A ⋃ B) = P(A)+ P(B), what does this mean? Explain each part.
7. Sally then creates the following diagram. Sally’s Venn diagram is incorrect. Why?
obabili 9.
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9. What is the probability that a randomly selected customer would order sourdough bread?
P(sourdough bread) =
10. What is the probability that a randomly selected customer would order sourdough bread
without cheese?
P(sourdough ∩ no cheese) = P(sourdough and no cheese) =
11. What is the probability that a person prefers wheat bread without cheese?
P(wheat ∩ no cheese) = P(wheat and no cheese) =
12. What is the estimated probability that a randomly selected customer would want their
sandwich with cheese?
P(sourdough cheese and wheat cheese) = P( ) =
13. If they serve 100 sandwiches at lunch on a particular day, how many orders with sourdough
should be prepared without cheese?
14. What is the probability that a randomly selected person would choose sourdough or without
cheese?
P(sourdough ∪ no cheese) = P(sourdough or no cheese) =
15. What is the probability that a randomly selected person would NOT choose sourdough or no
cheese?
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17. In a well-developed and carried out survey it was found that 4 out of 10 students have a pair of
sunglasses. How many students would you expect to have a pair of sunglasses out of a group of
45 students?
18. Data collected at a local mall indicted that 7 out of 20 men observed were wearing a hat. How
many would you expect to have been wearing hats if 7500 men were to be at the mall on a
similar day?
17
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9.4 Visualizing with Venn A Solidify Understanding Task
One of the attributes of Venn diagram’s is that it can be easy to see the
relationships within the data. In this task, we will create multiple Venn
diagrams using data and determine the events that create diagrams to
either have an intersection or for them to be mutually exclusive.
1. The following data represents the number of men and women passengers aboard
the titanic and whether or not they survived.
Survived Did not survive Total Men 146 659 805 Women 296 106 402 Total 442 765 1207
2. Create three Venn diagrams with this data.
a. Men vs Women
b. Women vs Survived
c. You choose the conditions
3. Create two probability statements using each of your Venn diagrams from question 2.
4. Create and label three different Venn diagrams using this data. Create at least one that
is mutually exclusive and at least one that has an intersection.
Sample size: 100
P(girl) = 42/100
P(girl or art) = (42/100 + 32/100)
P(art) = 3/100
P(not art) =
P(boy) =
5. Describe the conditions that create mutually exclusive Venn diagrams and those that
create intersections.
6. What conjecture can you make regarding the best way to create a Venn diagram from data
to highlight probabilities?
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Topic: Products of probabilities, multiplying and dividing fractions
Find the products or quotients below.
Set For each situation, one of the representations (two=way table, Venn diagram, tree diagram, context or probability notation) is provided. Use the provided information to complete the remaining representations.
9. Are you Blue?
Notation 24way Table
Key: Male = M Female = F Blue = B Not Blue = N
Sample size = 200
P(B) = 84/200 P(M) = 64/200
P(F|B) = 48/84 P(B|F) =
P(MB) = P(MB) =
(Continued on the next page)
obabili 9.
Blue Not Blue Total
Male
Female
Total
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(Continued from the last page)
Venn Diagram Tree Diagram
Write three observations you can make about this data.
10. Right and left handedness of a group.
Notation 2-way Table
Key: Male = M Female = F Lefty = L Righty = R
Sample size = 100 people
P(L) = P(M) =
P(F) = P(L|F) =
P(L|M) =
Venn Diagram Tree Diagram
Write three conditional statements regarding this data.
Lefty Righty Total
Male
Female
Total
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11. The most important meal of the day.
Notation 2-way Table
Key: Male = M Female = F Eats Breakfast = E Doesn’t Eat Breakfast = D
Sample size = P(E) = P(E|M) =
P(EM)= P(E|F) =
P(EF) = Venn Diagram Tree Diagram
Does this data surprise you? Why or why not.
Go Topic: Writing conditional statements from a two-way table.
14. Complete the table about eye color and write three conditional statements.
Blue Green Brown Other Total
Male 55 20 15 100
Female 20 10 Total 75 230
Biking Swimming Total
Male 50 Female 35 76
Total 85
Ice Cream Cake Total
Male 20 Female 10 60
Total 85
Eats Doesn’t Total
Male
Female
Total
685
E (58) =
M (62%)
D (42%) =
E (40%) =
F (38%)
D (60%) =
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9.5 Freddy Revisited A Solidify Understanding Task
Once Tyrell helped Freddy out in determining the amount and
type of food Freddy should prepare each day for his
restaurant, Freddy’s food waste decreased dramatically. Still,
Freddy noticed that during the week, he seemed to still have
more food prepared than he needed, and sometimes on the
weekend he would run out of something he needed. Tyrell
said another level of determining waste could be if Freddy
averaged the number of orders he received of fish and chicken
on a weekday and compared it to the average number of
orders he received of each on the weekend. Freddy thought
this was a good idea so started collecting data.
After two months, he had enough information to create a two way table representing the average
number of orders he received on the weekdays and on the weekends for fish and chicken. The data is
below:
Chicken Fish Total Weekday 65 79 144 Weekend 88 107 195 Total 153 186 339
1. What observations can you make? Explain to Freddy what this means (When does
Freddy seem to have the greatest business? Should he expect a greater percentage of
customers to order fish during the week or on the weekend? What else?)
2. Does the number of orders of chicken compared to fish depend on whether it is a
weekday or a weekend? What values from the table tell you this?
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Ready Topic: Quadratic Function Review
Set Topic: Independence
Determining the independence of events can sometimes be done by becoming familiar with the context in which the events occur and the nature of the events. There are also some ways of determining independence of events based on equivalent probabilities.
Two events, A and B, are independent if P (A and B) = P (A) ∙ P(B)
Additionally, two events, A and B, are independent if P(A|B) = = P(A)
Use these two ways of determining independent events to determine independence in the problems below and answer the questions.
obabili 9.
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Go Topic: Find probabilities from a two4way table.
The following data represents the number of men and women passengers aboard the titanic and
whether or not they survived.
Survived Did not survive Total
Men 146 659 805
Women 296 106 402
Total 442 765 1207
11. P(w) =
12. P(s) =
13. P(s|w) =
14. P(w or s) =
15. P(w or m) =
16. P(ns|w) =
17. P(m ∩ ns) =
obabili 9.
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9.6 Striving for Independence A Practice Understanding Task
Use your knowledge of conditional probability (the probability of A given B as P(A and B)/P(B) ) as well as the definition of independence: two events (A and B)) are said to be independent if P(A|B) = P(A) and P(B|A) = P(B ) to answer the following questions. Keep track of how you are determining independence for each type of representation.
1. Out of the 2000 students who attend a certain high school, 1400 students own cell
phones, 1000 own a tablet, and 800 have both. Suppose a student is randomly selected.
Create a Venn diagram model and use notation to answer the following questions.
a) What is the probability that a randomly selected student owns a cell phone?
b) What is the probability that a randomly selected students owns both a cell phone and
a tablet?
c) If a randomly selected student owns a cell phone (was one of the 1400 with a phone),
what is the probability that this student also owns a tablet?
d) How are questions c and d different?
e) Are the outcomes owns a cell phone and owns a tablet independent? Explain.
f) If question e is not independent, what number of students would own a tablet to
create independence?
2. Below is a partially completed tree diagram from the task Chocolate vs Vanilla.
a) Circle the parts of the diagram you would use to determine if choosing chocolate
is independent of being a male or female.
b) Complete the diagram so that choosing chocolate is independent of being male or
female.
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3. Use the titanic data below to answer the following questions.
Survived Did not survive Total Men 146 659 805 Women 296 106 402 Total 442 765 1207
a. Determine if survival was independent of gender. Explain.
b. If gender would not have mattered, what would have been the number of males that would
have survived, given the data for the number of females who survived and the total number of
passengers on the ship?
4. Determine whether the second scenario would be dependent or independent of the first
scenario. Explain.
a) Rolling a six-sided die and then drawing a card from a deck of 52 cards.
b) Drawing a card from a deck of 52 cards, and then drawing another card from the same deck.
c) Rolling a six-sided die, and then rolling it again.
d) Pulling a marble out of a bag, replacing it, and then pulling a marble out of the same bag.
e) Having 20 treats in five different flavors for a soccer team, with each player taking a
treat. 5. The definition of independence is that two events (A and B) are said to be independent if
P(A|B) = P(A) and P(B|A) = P(B)
Explain what this looks like in a Venn diagram, a tree diagram, and a two-way table.
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Ready Topic: End of year Review
http://www.flickr.com/photos/ronwls/
Solve each of the quadratics below using an appropriate method.
1. m2 + 15m + 56 = 0 2. 5x2 – 3x + 7 = 0
3. x2 − 10x + 21 = 0 4. 6x2 + 7x – 5 = 0
Set Topic: Representing Independent Events in Venn Diagrams
In each of the Venn Diagrams the number of outcomes for each event are given, use the provided information to determine the conditional probabilities or independence. The numbers in the Venn Diagram indicate the number of outcomes in that part of the sample space. 5.
a. How many total outcomes are possible?
b. P(A) =
c. P(B) =
d. P(A∩B) =
e. P(A|B) =
f. Are events A and B independent events? Why or why not?
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6. a. How many total outcomes are possible? b. P(E) = c. P(F) =P(E∩F) = d. P(E|F) =
e. Are events E and F independent events? Why or why not?
7. a. How many total outcomes are possible? b. P(X) = c. P(Y) = d. P(X∩Y) = e. P(X|Y) =
f. Are events X and Y independent events? Why or why not?
8. a. How many total outcomes are possible? b. P(K) = c. P(L) = d. P(K∩L) = e. P(K|L) =
f. Are events K and L independent events? Why or why not?
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Go Topic: Conditional Probability and Independence
Data gathered on the shopping patterns during the months of April and May of high school students from Peanut Village revealed the following. 38% of students purchased a new pair of shorts (call this event H), 15% of students purchased a new pair of sunglasses (call this event G) and 6% of students purchased both a pair of short and a pair of sunglasses.
9. Find the probability that a student purchased a pair of sunglasses given that you know they purchased a pair of shorts. P (G|H) =
10. Find the probability that a student purchased a pair of shorts or purchased a new pair of sunglasses. P (H ∪ G) =
11. Given the condition that you know a student has purchased at least one of the items. What is the probability that they purchased only one of the items?
12. Are the two events H and G independent of one another? Why or Why not? Given the data collected from 200 individuals concerning whether or not to extend the length of the school year in the table below answer the questions.
For Against No Opinion Youth (5 to 19) 7 35 12 Adults (20 to 55) 30 27 20 Seniors (55 +) 25 16 28 200
13. Given that condition that a person is an adult what is the probability that they are in favor of extending the school year? P(For|Adult) =
14. Given the condition that a person is against extending the school year what is the probability they are a Senior? P (Senior|Against) =
15. What is the probability that a person has no opinion given that they are a youth? P (no opinion|youth) =
Need Assistance? Check out these additional resources: http://www.youtube.com/watch?v=t6G8mL0w4xM
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Lesson 9: Learn how to Count! Factorials, Combination, Permutation Multiplication Counting Principle Many moons ago, at the dawn of the internet, companies scrambled to buy domain names. Let’s say your company wants to keep it simple and register a two character domain name (www.HI.com). How many possibilities for a domain are there if we just use letters? 26 letters
26*26 = 676 Example:
The Nom-Nom Ice Cream Shoppe sells:
32 flavors of delicious ice cream goodness 12 different toppings 3 different sizes
How many heavenly varieties of ordering are there?
32 * 12 * 3 = 1152 In general for a process involving multiple (say k) steps: there are n1 ways to do step 1 , n2 ways to do step 2, ..., and nk to do Step k
n1* n2*...* nk
How many possibilities for 3 main courses and 2 sides? 3 * 2 = 6
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Factorial How many different ways can we arrange 5 kittens? First: 5 possible Next: 4 possible Next: 3 possible Next: 2 possible Last: 1 possible 5*4*3*2*1 = 120 Multiplying by one less number for each iteration is called factorial 7*6*5*4*3*2*1 7! = 5040 n*(n-1)*(n-2)*…*1 = n! Example: Your bank issues you a bank card with a 4 digit pin. ____ _____ ____ ______ If you can repeat numbers how many possible PINs are there? 10 digits (0-9) so 10*10*10*10 = 104 = 10,000 If you can’t repeat numbers, how many possible PINs are there? 10 * 9 * 8 * 7 = 5040 OR
10!
6!=
10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1
6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1= 5040
WHAT?????
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Permutations and Combinations
The number of ways you can choose r objects in a specific order from n objects.
nPr = A teacher has been bribed to boost student grades to an A, B and C. He/She can't remember which 3 students' bribed him/her. How many different groups of 3 students can be made in a class room of 39? n = 39 r = 3 Why does order matter? A, B, C are different....
39!/36! = 39*38*37 = 54,834
The number of ways you can choose r objects from n objects.
nCr =nPr
r! nCr =
The r! in the denominator gets rid of the duplicates from when order did matter. A student has just won a trip for three people to Oahu for Spring Break. The student decides to take their BFFs. Alas, the student has 5 BFF's and can only choose two. How many possibilities are there?
5C2 = 5!
3!2!=
120
12 = 10
!
( )! !
n
n r r
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PERMUTATIONS and COMBINATIONS
If the order doesn't matter, it is a Combination. If the order does matter it is a
Permutation.
PRACTICE!
Determine whether each of the following situations is a Combination or
Permutation.
1. Creating an access code for a computer site using any 8 alphabet letters.
2. Determining how many different ways you can elect a Chairman and Co-Chairman of a
committee if you have 10 people to choose from.
3. Voting to allow 10 new members to join a club when there are 25 that would like to join.
4. Finding different ways to arrange a line-up for batters on a baseball team.
5. Choosing 3 toppings for a pizza if there are 9 choices.
Answers: 1. P 2. P 3. C 4. P 5. C
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Combinations:
Suppose that you can invite 3 friends to go with you to a concert. If you choose Jay,
Ted, and Ken, then this is no different from choosing Ted, Ken, and Jay. The order
that you choose the three names of your friends is not important. Hence, this is a
Combination problem.
Example Problem for Combination:
Suppose that you can invite 3 friends to go with you to a concert. You have 5 friends
that want to go, so you decide to write the 5 names on slips of paper and place
them in a bowl. Then you randomly choose 3 names from the bowl. If the five people
are Jay, Ted, Cal, Bob, and Ken, then write down all the possible ways that you
could choose a group of 3 people.
Here are all of the possible combinations of 3: Jay, Ted, Cal Ted, Cal, Bob Cal, Bob, Ken
Jay, Ted, Bob Ted, Cal, Ken
Jay, Ted, Ken Ted, Bob, Ken
Jay, Cal, Bob Jay, Cal,
Ken
Jay, Bob, Ken
PRACTICE: Go back to the beginning of this lesson and determine the number of
possible combinations for #3 and #5. Answers: 3,268,760 and 84 ,
respectively.
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Bob, Jay, Sue
Bob, Sue, Jay
Bob, Jay, Kim
Bob, Kim, Jay
Bob, Sue, Kim
Bob, Kim, Sue
Kim, Sue, Bob
Kim, Bob, Sue
Kim, Jay, Sue
Kim, Sue, Jay
Kim, Jay, Bob
Kim, Bob, Jay
Sue, Kim, Bob
Sue, Bob, Kim
Sue, Jay, Kim
Sue, Kim, Jay
Sue, Jay, Bob
Sue, Bob, Jay
Jay, Sue, Kim
Jay, Kim, Sue
Jay, Sue, Bob,
Jay, Bob, Sue
Jay, Kim, Bob
Jay, Bob, Kim
Permutations:
A permutation is used when re-arranging the elements of the set creates a new situation.
Example Problem for Permutation:
How many ways could we get 1st, 2nd, and 3rd place winners in a race with the
following 4 people? Jay, Sue, Kim, and Bob
**Note: since winning first place is different than winning second place, the set
{Jay, Sue, Kim} would mean something different than {Jay, Kim, Sue}.
Here is why they are different:
{Jay, Sue, Kim} indicates that Jay won 1st , Sue won 2nd, and Kim won 3rd .
{Jay, Kim, Sue} indicates that Jay won 1st, Kim won 2nd, and Sue won
3rd Here are all of the possibilities for 1st, 2nd, 3rd in that order:
There are 24 permutations.
Another way to determine the number of permutations is to use the following formula:
PRACTICE: Go back to the beginning of this lesson and determine the number of possible permutations for #1 and #2.
Answers: 6.299 ∗ 1010 and 90 , respectively.
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MORE COMBINATION AND PERMUTATION PRACTICE PROBLEMS:
1. Suppose that 7 people enter a swim meet. Assuming that there are no ties, in how many ways could the gold,
silver, and bronze medals be awarded?
2. How many different committees of 3 people can be chosen to work on a special project from a group of 9
people?
3. A coach must choose how to line up his five starters from a team of 12 players. How many different ways
can the coach choose the starters?
4. John bought a machine to make fresh juice. He has five different fruits: strawberries, oranges, apples,
pineapples, and lemons. If he only uses two fruits, how many different juice drinks can John make?
5. How many different four-letter passwords can be created for a software access if no letter can be used
more than once?
6. How many different ways you can elect a Chairman and Co-Chairman of a committee if you have 10 people
to choose from.
7. There are 25 people who work in an office together. Five of these people are selected to go together to the
same conference in Orlando, Florida. How many ways can they choose this team of five people to go to the
conference?
8. There are 25 people who work in an office together. Five of these people are selected to attend five different
conferences. The first person selected will go to a conference in Hawaii, the second will go to New York, the
third will go to San Diego, the fourth will go to Atlanta, and the fifth will go to Nashville. How many such
selections are possible?
9. John couldn’t recall the Serial number on his expensive bicycle. He remembered that there were 6 different
digits, none used more than once, but couldn’t remember what digits were used. He decided to write down all of
the possible 6 digit numbers . How many different possibilities will he have to create?
10. How many different 7-card hands can be chosen from a standard 52-card deck?
11. One hundred twelve people bought raffle tickets to enter a random drawing for three prizes. How many
ways can three names be drawn for first prize, second prize, and third prize?
12. A disc jockey has to choose three songs for the last few minutes of his evening show. If there are nine
songs that he feels are appropriate for that time slot, then how many ways can he choose and arrange to play
three of those nine songs?
Answers:
1. 7P3 = 210, 2. 9C3 = 84, 3. 12P5 = 95,040, 4. 5C2 = 10, 5. 26P4 = 358,800, 6.
10P2 = 90, 7. 25C5 = 53,130, 8. 25P5 = 6,375,600, 9. 10P6 = 151,200, 10. 52C7 =
133,784,560, 11. 112P3 = 1,367,520, 12. 9P3 = 504
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Tutor-USA.com Worksheet Name: Permutations Date: Period: Score:
Determine the value:
1) How many numbers of 3 different digits can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9?
2) How many numbers of at most 3 different digits each can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9?
3) In how many ways can 4 boys and 3 girls be seated in a row containing seven seats a) If they sit anywhere.
b) If the girls and boys must alternate.
4) A baseball manager insists on having his best hitter bat fourth and the pitcher bat last. How many batting orders are possible?
5) A language teacher wants to keep books of the same language together on his shelf. If he has 12 spaces for 5 French, 4 Italian, and 3 German books, in how many ways can they be placed on his shelf?
6) In how many ways can 8 people be seated around a table if 2 people insist of sitting next to
each other?
7) How many license plates can be made using any two letters for the first 2 places and any of the numbers 0 through 9 for the last 3 given that no letter or number is repeated?
8) How many permutations are there of the letters of the word “Tennessee”?
9) In how many ways can 7 different colored beads be made into a bracelet?
10) Solve: P(n, 5) = 20P(n, 3) for n.
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Finding Probabilities Using Combinations and Permutations Combinations can be used in finding probabilities as illustrated in the next example. Example Given a class of 12 girls and 10 boys. (a) In how many ways can a committee of five consisting of 3 girls and 2 boys be chosen? (b) What is the probability that a committee of five, chosen at random from the class, consists of three girls and two boys? (c) How many of the possible committees of five have no boys?(i.e. consists only of girls) (d) What is the probability that a committee of five, chosen at random from the class, consists only of girls?
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Practice Problems
1. John and Beth are hoping to be selected from their class of 30 as president and vice-president of the Social Committee. If the three-person committee (president, vice-president, and secretary) is selected at random, what is the probability that John and Beth would be president and vice president?
2. There are 10 boys and 13 girls in Mr. Benson’s fourth-grade class and 12 boys and 11 girls in Mr. Johnson fourth-grade class. A picnic committee of six people is selected at random from the total group of students in both classes.
a. What is the probability that all the committee members are girls?
b. What is the probability that the committee has three girls and three boys?
3. A school dance committee of 4 people is selected at random from a group of 6 ninth graders, 11 eighth graders, and 10 seventh graders.
a. What is the probability that the committee has all seventh graders?
b. What is the probability that the committee has no seventh graders?
4. In an effort to promote school spirit, Georgetown High School created ID numbers with just the letters G, H, and S. If each letter is used exactly three times,
a. How many nine-letter ID numbers can be generated?
b. What is the probability that a random ID number starts with GHS?
5. The license plates in the state of Utah consist of three letters followed by three single-digit numbers. a. If Edward’s initials are EAM, what is the probability that his license plate will have his initials on
it (in any order)?
b. What is the probability that his license plate will have his initials in the correct order?
