Unit 1: Name: ______________________Probability CCSS.Math.Content.7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. CCSS.Math.Content.7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. CCSS.Math.Content.7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. CCSS.Math.Content.7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. CCSS.Math.Content.7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. CCSS.Math.Content.7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. CCSS.Math.Content.7.SP.C.8a-c Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. CCSS.Math.Content.8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Date Topic Test/QuizThursday, January 22 Basic ProbabilityFriday, January 23 Independent and Dependent EventsMonday, January 26 Mutually Exclusive vs Not Mutually ExclusiveTuesday, January 27 Review/Flex QuizWednesday, January 28 Conditional ProbabilityThursday, January 29 Permutations and CombinationsFriday, January 30 Theoretical vs Experimental 1

Monday, February 2 ReviewTuesday, February 3 Test Test

2

Factors of 12

16 12 2

4

Factors of

168

16

Guided Notes: Sample Spaces, Subsets, and Basic ProbabilitySample Space: __________________________________________________________________________________________________________________________________________________________________________List the sample space, S, for each of the following:

a. Tossing a coin: b. Rolling a six-sided die:c. Drawing a marble from a bag that contains two red, three blue, and one white marble:

Intersection of two sets (A B): __________________________________________________________________________________________________________________________________________________________________________Union of two sets (A B): __________________________________________________________________________________________________________________________________________________________________________Example: Given the following sets, find A B and A B

A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15}A B = ____________________ A B = _____________________

Venn Diagram: __________________________________________________________________________________________________________________________________________________________________________Picture:

Example: Use the Venn Diagram to answer the following questions:1. What are the elements of set A?

2. What are the elements of set B?

3. Why are 1, 2, and 4 in both sets?

4. What is A B?

5. What is A B?

Example: In a class of 60 students, 21 sign up for chorus, 29 sign up for band, and 5 take both. 15 students in the class are not enrolled in either band or chorus.

6. Put this information into a Venn Diagram. If the sample space, S, is the set of all students in the class, let students in chorus be set A and students in band be set B.

7. What is A B? ______________________

8. What is A B? _____________________

Compliment of a set: ____________________________________________________________________________________________________________________________________________________________________________________________________

3

• Ex: S = {…-3,-2,-1,0,1,2,3,4,…}A = {…-2,0,2,4,…}If A is a subset of S, what is AC? ________________________

Example: Use the Venn Diagram above to find the following:9. What is AC? _______________ BC? ______________

10. What is (A B)C? ______________________

11. What is (A B)C? ______________________

Basic ProbabilityProbability of an Event: P(E) = _______________________________Note that P(AC) is every outcome except (or not) A, so we can find P(AC) by finding _______________________.Why do you think this works? __________________________________________________________________Example: An experiment consists of tossing three coins.

12. List the sample space for the outcomes of the experiment.__________________________________________________________________

13. Find the following probabilities:a. P(all heads) _________________b. P(two tails) _________________c. P(no heads) _________________

d. P(at least one tail) _____________

e. How could you use compliments to find d? __________________________________________________________________________________________________________________________________________________________________________

Example: A bag contains six red marbles, four blue marbles, two yellow marbles and 3 white marbles. One marble is drawn at random.

14. List the sample space for this experiment. _____________________________________________________15. Find the following probabilities:

a. P(red) ____________b. P(blue or white) ____________c. P(not yellow) ____________

Note that we could either count all the outcomes that are not yellow or we could think of this as being 1 – P(yellow). Why is this? ______________________________________________________________________________________________________________________________________________________________________________________

Example: A card is drawn at random from a standard deck of cards. Find each of the following:16. P(heart) ____________17. P(black card) ____________18. P(2 or jack) ____________19. P(not a heart) ____________

Odds: The odds of an event occurring are equal to the ratio of ________________ to __________________. 4

Odds = _________________________________________________

20. The weather forecast for Saturday says there is a 75% chance of rain. What are the odds that it will rain on Saturday?

• What does the 75% in this problem mean?

• The favorable outcome in this problem is that it rains:

• Odds(rain) =

• Should you make outdoor plans for Saturday?

21. What are the odds of drawing an ace at random from a standard deck of cards?

PracticeOrganize the data into the circles. Factors of 64: 1, 2, 4, 8, 16, 32, 64 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Answer Questions about the diagram below

1) How many students play sports year-round?2) How many students play sports in the spring and fall?3) How many students play sports in the winter and fall?4) How many students play sports in the winter and spring?5) How many students play only one sport? 5

Fall Sports Winter Sports21

13 8

6 3 2

19

6) How many students play at least two sports?

7) Suppose you have a standard deck of 52 cards. Let:

a. Describe for this experiment, and find the probability of . b. Describe for this experiment, and find the probability of .

8) Suppose a box contains three balls, one red, one blue, and one white. One ball is selected, its color is observed, and then the ball is placed back in the box. The balls are scrambled, and again, a ball is selected and its color is observed. What is the sample space of the experiment?

9) Suppose you have a jar of candies: 4 red, 5 purple and 7 green. Find the following probabilities of the following events: Selecting a red candy. Selecting a purple candy. Selecting a green or red candy.Selecting a yellow candy. Selecting any color except a green candy.Find the odds of selecting a red candy.Find the odds of selecting a purple or green candy.

10) What is the sample space for a single spin of a spinner with red, blue, yellow and green sections spinner? What is the sample space for 2 spins of the first spinner? If the spinner is equally likely to land on each color, what is the probability of landing on red in one spin? What is the probability of landing on a primary color in one spin?What is the probability of landing on green both times in two spins?

11) Consider the throw of a die experiment. Assume we define the following events:

Describe for this experiment. Describe for this experiment.

Calculate and , assuming the die is fair.

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Guided Notes: Probability of Independent and Dependent Events

Independent Events: __________________________________________________________________________________________________________________________________________________________________________

Dependent Events: __________________________________________________________________________________________________________________________________________________________________________

Suppose a die is rolled and then a coin is tossed.

• Explain why these events are independent. ______________________________________________________________________________

Fill in the table to describe the sample space:

Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6

Head

Tail

• How many outcomes are there for rolling the die? _________________

• How many outcomes are there for tossing the coin? ________________

• How many outcomes are there in the sample space of rolling the die and tossing the coin? ____________

• Is there another way to decide how many outcomes are in the sample space? ________________________________________________________________________________________

Let’s see if this works for another situation.

A fast food restaurant offers 5 sandwiches and 3 sides. How many different meals of a sandwich and side can you order?

• If our theory holds true, how could we find the number of outcomes in the sample space? _________________________________________________________

7

• Make a table to see if this is correct.

Were we correct? _____________

Probabilities of Independent Events

The probability of independent events is__________________________________________________________, denoted by ___________________________________________________.

Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6

Head

Tail

Fill in the table again and then use the table to find the following probabilities:

1. P(rolling a 3) = _____________________________________

2. P(Tails) = _____________________________________

3. P(rolling a 3 AND getting tails) = _____________________________________

4. P(rolling an even) = _____________________________________

5. P(heads) = _____________________________________

6. P(rolling an even AND getting heads) = _____________________________________

What do you notice about the answers to 3 and 6?

____________________________________________________________________________________________________________________________________________________________________________________________________

Multiplication Rule of Probability

8

• The probability of two independent events occurring can be found by the following formula:

__________________________________________________________________________

Examples:

1. At City High School, 30% of students have part-time jobs and 25% of students are on the honor roll. What is the probability that a student chosen at random has a part-time job and is on the honor roll? Write your answer in context.

2. The following table represents data collected from a grade 12 class in DEF High School.

Suppose 1 student was chosen at random from the grade 12 class.

(a) What is the probability that the student is female? ________________________________

(b) What is the probability that the student is going to university?______________________________

Now suppose 2 people both randomly chose 1 student from the grade 12 class. Assume that it's possible for them to choose the same student.

(c) What is the probability that the first person chooses a student who is female and the second person chooses a student who is going to university? ____________________________________________________________

3. Suppose a card is chosen at random from a deck of cards, replaced, and then a second card is chosen.Would these events be independent? How do we know? __________________________________________________________________________________________What is the probability that both cards are 7s?

__________________________________________________________________________________________

Probabilities of Depended Events

Determine whether the events are independent or dependent:

1. Selecting a marble from a container and selecting a jack from a deck of cards. ____________________

2. Rolling a number less than 4 on a die and rolling a number that is even on a second die. ____________________9

3. Choosing a jack from a deck of cards and choosing another jack, without replacement. ____________________

4. Winning a hockey game and scoring a goal. ____________________

• We cannot use the multiplication rule for finding probabilities of dependent events because the one event affects the probability of the other event occurring.

• Instead, we need to think about how the occurrence of one event will effect the sample space of the second event to determine the probability of the second event occurring.

• Then we can multiply the new probabilities.

Examples:

1. Suppose a card is chosen at random from a deck, the card is NOT replaced, and then a second card is chosen from the same deck. What is the probability that both will be 7s?

• This is similar the earlier example, but these events are dependent? How do we know? ________________________________________________________________________

• How does the first event affect the sample space of the second event? ________________________________________________________________________

Now find the probability that both cards will be 7s.

2. A box contains 5 red marbles and 5 purple marbles. What is the probability of drawing 2 purple marbles and 1 red marble in succession without replacement?

3. In Example 2, what is the probability of first drawing all 5 red marbles in succession and then drawing all 5 purple marbles in succession without replacement?

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Independent and Dependent Events

1. Determine which of the following are examples of independent or dependent events.

a. Rolling a 5 on one die and rolling a 5 on a second die.

b. Choosing a cookie from the cookie jar and choosing a jack from a deck of cards.

c. Selecting a book from the library and selecting a book that is a mystery novel.

d. Choosing an 8 from a deck of cards, replacing it, and choosing a face card.

e. Choosing a jack from a deck of cards and choosing another jack, without replacement.

2. A coin and a die are tossed. Calculate the probability of getting tails and a 5.

3. In Tania's homeroom class, 9% of the students were born in March and 40% of the students have a blood

type of O+. What is the probability of a student chosen at random from Tania's homeroom class being born

in March and having a blood type of O+?

4. If a baseball player gets a hit in 31% of his at-bats, what it the probability that the baseball player will get a

hit in 5 at-bats in a row?

5. What is the probability of tossing 2 coins one after the other and getting 1 head and 1 tail?

6. 2 cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is

the probability that they both will be clubs?

7. 2 cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is

the probability that they both will be face cards?

8. If the probability of receiving at least 1 piece of mail on any particular day is 22%, what is the probability of

not receiving any mail for 3 days in a row?

11

9. Jonathan is rolling 2 dice and needs to roll an 11 to win the game he is playing. What is the probability that

Jonathan wins the game?

10. Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly

beans. What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean

and then reaching in again and pulling out a red jelly bean? Assume that the first jelly bean is not replaced.

11. For question 10, what if the order was reversed? In other words, what is the probability of Thomas

reaching into the bag and pulling out a red jelly bean and then reaching in again and pulling out a blue or

green jelly bean without replacement?

12. What is the probability of drawing 2 face cards one after the other from a standard deck of cards without

replacement?

13. There are 3 quarters, 7 dimes, 13 nickels, and 27 pennies in Jonah's piggy bank. If Jonah chooses 2 of the

coins at random one after the other, what is the probability that the first coin chosen is a nickel and the

second coin chosen is a quarter? Assume that the first coin is not replaced.

14. For question 13, what is the probability that neither of the 2 coins that Jonah chooses are dimes? Assume

that the first coin is not replaced.

15. Jenny bought a half-dozen doughnuts, and she plans to randomly select 1 doughnut each morning and eat

it for breakfast until all the doughnuts are gone. If there are 3 glazed, 1 jelly, and 2 plain doughnuts, what is

the probability that the last doughnut Jenny eats is a jelly doughnut?

16. Steve will draw 2 cards one after the other from a standard deck of cards without replacement. What is the

probability that his 2 cards will consist of a heart and a diamond?

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Guided Notes: Mutually Exclusive and Inclusive Events

Mutually Exclusive Events

Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or you roll a 2?

• Can these both occur at the same time? Why or why not? ______________________________________________________________________________________________________________________________________________________________________________________

Mutually Exclusive Events:

____________________________________________________________________________________________________________________________________________________________________________________________________

• The probability of two mutually exclusive events occurring at the same time, P(A and B), is ______________

To find the probability of one of two mutually exclusive events occurring, use the following formula:

__________________________________________________________________________________________________

Examples:

1. If you randomly chose one of the integers 1 – 10, what is the probability of choosing either an odd number or an even number?

Are these mutually exclusive events? Why or why not? ______________________________________________

Complete the following statement:

P(odd or even) = P(_____) + P(_____)

Now fill in with numbers:

P(odd or even) = _______ + ________ = _______________________________________________

Does this answer make sense? _________________________________________________________________

2. Two fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal to 10?Are these events mutually exclusive? ____________________________________________________________Sometimes using a table of outcomes is useful. Complete the following table using the sums of two dice:

1 2 3 4 5 61 2 3 4 5 6 72 3 43 4456 13

P(getting a sum less than 7 OR sum of 10) = ________________________________________________________This means __________________________________________________________________________________

Mutually Inclusive Events

Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or a number less than 4?

• Can these both occur at the same time? If so, when? ________________________________________________________________________________________________________________________________________________________________________________

Mutually Inclusive Events: ____________________________________________________________________________________________________________________________________________________________________________________________________

Probability of the Union of Two Events: The Addition Rule:

__________________________________________________________________________________________________

***____________________________________________________________________________________________***

Examples:

1. What is the probability of choosing a card from a deck of cards that is a club or a ten?P(choosing a club or a ten) =

2. What is the probability of choosing a number from 1 to 10 that is less than 5 or odd?

3. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the first 10 letters of the alphabet on it or randomly choosing a tile with a vowel on it?

4. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the last 5 letters of the alphabet on it or randomly choosing a tile with a vowel on it?

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Name___________________ CCM2 Unit 6 Lesson 2 Homework 1

Mutually Exclusive and Inclusive Events

1. 2 dice are tossed. What is the probability of obtaining a sum equal to 6?

2. 2 dice are tossed. What is the probability of obtaining a sum less than 6?

3. 2 dice are tossed. What is the probability of obtaining a sum of at least 6?

4. Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly beans.

What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean?

5. A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart or

spade? Are these events mutually exclusive?

6. 3 coins are tossed simultaneously. What is the probability of getting 3 heads or 3 tails? Are these events mutually

exclusive?

7. In question 6, what is the probability of getting 3 heads and 3 tails when tossing the 3 coins simultaneously?

8. Are randomly choosing a person who is left-handed and randomly choosing a person who is right-handed mutually

exclusive events? Explain your answer.

9. Suppose 2 events are mutually exclusive events. If one of the events is randomly choosing a boy from the freshman

class of a high school, what could the other event be? Explain your answer.

10. Consider a sample set as . Event is the multiples of 4, while event is

the multiples of 5. What is the probability that a number chosen at random will be from both and ? 15

11. For question 10, what is the probability that a number chosen at random will be from either or ?

12. Jack is a student in Bluenose High School. He noticed that a lot of the students in his math class were also in his

chemistry class. In fact, of the 60 students in his grade, 28 students were in his math class, 32 students were in his

chemistry class, and 15 students were in both his math class and his chemistry class. He decided to calculate what

the probability was of selecting a student at random who was either in his math class or his chemistry class, but not

both. Draw a Venn diagram and help Jack with his calculation.

13. Brenda did a survey of the students in her classes about whether they liked to get a candy bar or a new math pencil

as their reward for positive behavior. She asked all 71 students she taught, and 32 said they would like a candy bar,

25 said they wanted a new pencil, and 4 said they wanted both. If Brenda were to select a student at random from

her classes, what is the probability that the student chosen would want:

1. a candy bar or a pencil?

2. neither a candy bar nor a pencil?

14. A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart

or a face card? Are these events mutually inclusive?

15. What is the probability of choosing a number from 1 to 10 that is greater than 5 or even?

16. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of

reaching into the bag and randomly choosing a tile with one of the letters in the word ENGLISH on it or randomly

choosing a tile with a vowel on it?

17. Are randomly choosing a teacher and randomly choosing a father mutually inclusive events? Explain your

answer.

18. Suppose 2 events are mutually inclusive events. If one of the events is passing a test, what could the other event

be? Explain your answer.

16

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Guided Notes: Conditional Probability

Conditional Probability: -____________________________________________________________________________________________________________________________________________________________________________________________________

Examples of conditional probability:

The conditional probability of A given B is expressed as ___________________________

The formula is: ____________________________________________________________

Examples of Conditional Probability:

1. You are playing a game of cards where the winner is determined by drawing two cards of the same suit. What is the probability of drawing clubs on the second draw if the first card drawn is a club?

2. A bag contains 6 blue marbles and 2 brown marbles. One marble is randomly drawn and discarded. Then a second marble is drawn. Find the probability that the second marble is brown given that the first marble drawn was blue.

3. In Mr. Jonas' homeroom, 70% of the students have brown hair, 25% have brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes?

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Using Two-Way Frequency Tables to Compute Conditional Probabilities

1. Suppose we survey all the students at school and ask them how they get to school and also what grade they are in. The chart below gives the results. Complete the two-way frequency table:

Bus Walk Car Other Total9th or 10th 106 30 70 411th or 12th 41 58 184 7Total

Suppose we randomly select one student.

a. What is the probability that the student walked to school?

b. P(9th or 10th grader)

c. P(rode the bus OR 11th or 12th grader)

d. What is the probability that a student is in 11th or 12th grade given that they rode in a car to school?

e. What is P(Walk|9th or 10th grade)?

2. The manager of an ice cream shop is curious as to which customers are buying certain flavors of ice cream. He decides to track whether the customer is an adult or a child and whether they order vanilla ice cream or chocolate ice cream. He finds that of his 224 customers in one week that 146 ordered chocolate. He also finds that 52 of his 93 adult customers ordered vanilla. Build a two-way frequency table that tracks the type of customer and type of ice cream.

Vanilla Chocolate TotalAdultChildTotal

a. Find P(vanilla|adult)

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b. Find P(child|chocolate)

3. A survey asked students which types of music they listen to? Out of 200 students, 75 indicated pop music and 45 indicated country music with 22 of these students indicating they listened to both. Use a Venn diagram to find the probability that a randomly selected student listens to pop music given that they listen country music.

Using Conditional Probability to Determine if Events are Independent

If two events are statistically independent of each other, then:

________________________________________________________________________________________________

Let’s revisit some previous examples and decide if the events are independent.

1. You are playing a game of cards where the winner is determined by drawing two cards of the same suit without replacement. What is the probability of drawing clubs on the second draw if the first card drawn is a club?

Are the two events independent? Let drawing the first club be event A and drawing the second club be event B.

2. You are playing a game of cards where the winner is determined by drawing tow cards of the same suit. Each player draws a card, looks at it, then replaces the card randomly in the deck. Then they draw a second card.

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What is the probability of drawing clubs on the second draw if the first card drawn is a club? Are the two events independent?

3. In Mr. Jonas' homeroom, 70% of the students have brown hair, 25% have brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes?

Are event A, having brown hair, and event B, having brown eyes, independent?

4. Using the table from the ice cream shop problem, determine whether age and choice of ice cream are independent events.

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Name___________________ CCM2 Unit 6 Lesson 2 Homework 1

Conditional Probability

1. Compete the following table using sums from rolling two dice. Us e the table to answer questions 2-5.

1 2 3 4 5 6123456

2. 2 fair dice are rolled. What is the probability that the sum is even given that the first die that is rolled is a 2?

3. 2 fair dice are rolled. What is the probability that the sum is even given that the first die rolled is a 5?

4. 2 fair dice are rolled. What is the probability that the sum is odd given that the first die rolled is a 5?

5. Steve and Scott are playing a game of cards with a standard deck of playing cards. Steve deals Scott a black king.

What is the probability that Scott’s second card will be a red card?

6. Sandra and Karen are playing a game of cards with a standard deck of playing cards. Sandra deals Karen a red seven.

What is the probability that Karen’s second card will be a black card?

7. Donna discusses with her parents the idea that she should get an allowance. She says that in her class, 55% of her

classmates receive an allowance for doing chores, and 25% get an allowance for doing chores and are good to their

parents. Her mom asks Donna what the probability is that a classmate will be good to his or her parents given that

he or she receives an allowance for doing chores. What should Donna's answer be?

8. At a local high school, the probability that a student speaks English and French is 15%. The probability that a student

speaks French is 45%. What is the probability that a student speaks English, given that the student speaks French?

9. On a game show, there are 16 questions: 8 easy, 5 medium-hard, and 3 hard. If contestants are given questions

randomly, what is the probability that the first two contestants will get easy questions?

10. On the game show above, what is the probability that the first contestant will get an easy question and the second

contestant will get a hard question?

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11. Figure 2.2 shows the counts of earned degrees for several colleges on the East Coast. The level of degree and the gender of the degree recipient were tracked. Row & Column totals are included.

a. What is the probability that a randomly selected degree recipient is a female? b. What is the probability that a randomly chosen degree recipient is a man? c. What is the probability that a randomly selected degree recipient is a woman, given that they

received a Master's Degree? d. For a randomly selected degree recipient, what is P(Bachelor's Degree|Male)?

12. Animals on the endangered species list are given in the table below by type of animal and whether it is domestic or foreign to the United States. Complete the table and answer the following questions.

Mammals Birds Reptiles Amphibians Total

United States 63 78 14 10

Foreign 251 175 64 8

Total

An endangered animal is selected at random. What is the probability that it is: a. a bird found in the United States?

b. foreign or a mammal?

c. a bird given that it is found in the United States?

d. a bird given that it is foreign?

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Guided Notes: Permutations and Combinations

Fundamental Counting Principle: ____________________________________________________________________________________________________________________________________________________________________________________________________

Example: A student is to roll a die and flip a coin. How many possible outcomes will there be?

Example: For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from?

Permutation: ____________________________________________________________________________________________________________________________________________________________________________________________________

Example: Find the number of ways to arrange the letters ABC:

To find the number of Permutations of n items chosen r at a time, you can use the formula for finding P(n,r) or nPr :

Example: A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

24

You can use your calculator to find permutations:

Example: From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

Combination: ____________________________________________________________________________________________________________________________________________________________________________________________________

To find the number of Combinations of n items chosen r at a time, C(n,r) or nCr, you can use the formula:

Example: To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

You can use your calculator to find combinations:

25

Example: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

Example: A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards?

Example: The 25-member senior class council is selecting officers for president, vice president and secretary. Emily would like to be president, David would like to be vice president, and Jenna would like to be secretary. If the offices are filled at random, beginning with president, what is the probability that they are selected for these offices?

Example: The 25-member senior class council is selecting members for the prom committee. Stephen, Marcus and Sabrina want would like to be on this committee. If the members are selected at random, what is the probability that all three are selected for this committee?

26

Name___________________ CCM2 Unit 6 Lesson 2 Homework 2Permutations and Combinations

For 1-5, find the number of permutations or combinations. 1.

2.

3.

4.

5.

6.

For 7-20, identify the following as Permutations, Combinations or Counting Principle problems (Do not solve)

7. In a race in which six automobiles are entered and there are not ties, in how many ways can the first four finishers come in?

8. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

9. A book club offers a choice of 8 books from a list of 40. In how many ways can a member make a collection?

10. A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

11. From a club of 20 people, in how many ways can a group of three members be selected?

12. From the 30 pictures I have of my daughter’s first birthday, my digital picture frame will only hold 3 at a time. a. How many different groups of 3 pictures can I put on the frame?

b. What if I just wanted to fill the first three places with my favorite, best smile and best smashing of the cake?

13. A popular brand of pen is available in three colors (red, green or blue) and four tips (bold, medium, fine or micro). How many different choices of pens do you have with this brand?

14. A corporation has ten members on its board of directors. In how many ways can it elect a president, vice-president, secretary and treasurer?

15. For a segment of a radio show, a disc jockey can play 7 songs. If there are 12 songs to select from, in how many ways can the program for this segment be arranged?

16. How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

17. What if the director in #28 wanted to fill positions of lead, supporting actor, extra 1 and extra 2?

18. From the 20 CD’s you bought this past year, you plan to take 3 with you on vacation. How many different sets of three CD’s can you take?

19. Suppose you find 7 articles related to the topic of your research paper. In how many ways can you choose 5 articles to read?

20. You want to get a cell phone and you must decide on the right plan. If there are 10 different phones, 6 different calling plans and 3 different texting plans, how many different plans could you pick from if you can choose one phone, one calling plan and one texting plan?

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For 21-34 calculate the number of possibilities.

21. The ski club with ten members is to choose three officers captain, co-captain & secretary, how many ways can those offices be filled?

22. The company Sea Esta has ten members on its board of directors. In how many different ways can it elect a president, vice-president, secretary and treasurer?

23. For a segment of a radio show, a disc jockey (Dr. Jams) can play 4 songs. If there are 8 to select from, in how many ways can the program for this segment be arranged?

24. Suppose you are asked to list, in order or preference, the three best movies you have seen this year. If you saw 10 movies during the year, in how many ways can the three best be chosen and ranked?

25. In the Long Beach Air Race six planes are entered and there are no ties, in how many ways can the first three finishers come in?

26. In a production of Grease, eight actors are considered for the male roles of Danny, Kenickie, and Marty. In how many ways can the director cast the male roles?

27. Seven bands have volunteered to perform at a benefit concert, but there is only enough time for four of the bands to play. How many lineups are possible?

28. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

29. A four-person committee is to be elected from an organization’s membership of 11 people. How many different committees are possible?

30. You are on your way to Hawaii (Aloha) and of 15 possible books your parents say you can only take 10. How many different collections of 10 books can you take?

31. There are 12 standbys who hope to get on your flight to Hawaii, but only 6 seats are available on the plane. How many different ways can the 6 people be selected?

32. To win the small county lottery, one must correctly select 3 numbers from 30 numbers. The order in which the selection is made does not matter. How many different selections are possible?

33. How many ways can you plant a rose bush, a lavender bush and a hydrangea bush in a row?

34. How many ways can you pick a president, a vice president, a secretary and a treasurer out of 28 people for student council?

For 35-37, find the probabilities. 35. What is the probability that a randomly generated arrangement of the letters A,E,L, Q and U will result in spelling

the word EQUAL? 36. A prepaid telephone calling card comes with a randomly selected 4-digit PIN, using the digits 1 through 9 without

repeating any digits. What is the probability that the PIN for a card chosen at random does not contain the number 7?

37. A town lottery requires players to choose three different numbers from the numbers 1 through 36. What is the probability that a player’s numbers match all three numbers chosen by the computer?

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Investigation: Theoretical vs. Experimental Probability

Part 1: Theoretical ProbabilityProbability is the chance or likelihood of an event occurring. We will study two types of probability, theoretical and experimental.

Theoretical Probability: the probability of an event is the ratio or the number of favorable outcomes to the total possible outcomes.

P(Event) = Number or favorable outcomes Total possible outcomes

Sample Space: The set of all possible outcomes. For example, the sample space of tossing a coin is {Heads, Tails} because these are the only two possible outcomes. Theoretical probability is based on the set of all possible outcomes, or the sample space.

1. List the sample space for rolling a six-sided die (remember you are listing a set, so you should use brackets {} ):

Find the following probabilities:P(2) P(3 or 6) P(odd) P(not a 4)

P(1,2,3,4,5, or 6) P(8)

2. List the sample space for tossing two coins:

Find the following probabilities:P(two heads) P(one head and one tail) P(head, then tail)

P(all tails) P(no tails)

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3. Complete the sample space for tossing two six-sided dice:{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),(2,1), (2,2), (2,__), ____, ____, ____,(3,1), ____, ____, ____, ____, ____,____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____}

Find the following probabilities:P(a 1 and a 4) P(a 1, then a 4) P(sum of 8)

P(sum of 12) P(doubles) P(sum of 15)

4. When would you expect the probability of an event occurring to be 1, or 100%? Describe an event whose probability of occurring is 1.

5. When would you expect the probability of an event occurring to be 0, or 0%? Describe an event whose probability of occurring is 0.

Part 2: Experimental ProbabilityExperimental Probability: the ratio of the number of times the event occurs to the total number of trials.

P(Event) = Number or times the event occurs Total number of trials

1. Do you think that theoretical and experimental probabilities will be the same for a certain event occurring? Explain your answer.

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2. Roll a six-sided die and record the number on the die. Repeat this 9 more times

Number on Die Tally Frequency123456Total 10

Based on your data, find the following experimental probabilities:

P(2) P(3 or 6) P(odd) P(not a 4)

How do these compare to the theoretical probabilities in Part 1? Why do you think they are the same or different?

3. Record your data on the board (number on die and frequency only). Compare your data with other groups in your class. Explain what you observe about your data compared to the other groups. Try to make at least two observations.

4. Combine the frequencies of all the groups in your class with your data and complete the following table:

Number on Die Frequency123456Total

Based on the whole class data, find the following experimental probabilities:

P(2) P(3 or 6) P(odd) P(not a 4)

How do these compare to your group’s probabilities? How do these compare to the theoretical probabilities from Part 1?

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What do you think would happen to the experimental probabilities if there were 200 trials? 500 trials? 1000 trials? 1,000,000 trials?

5. On your graphing calculator, go to APPS and open Prob Sim. Press any key and then select 2: Roll dice.

Click Roll. Notice that there will be a bar on the graph at the right. What does this represent?

Now push +1 nine more times. Push the right arrow to see the frequency of each number on the die. How many times did you get a 1?______ A 2?________ A 5?

Now press the +1, +10, and +50 buttons until you have rolled 100 times. Based on the data, find the following experimental probabilities:

P(2) P(3 or 6) P(odd) P(not a 4)

Press the +50 button until you have rolled 1000 times. Based on the data, find the following experimental probabilities:

P(2) P(3 or 6) P(odd) P(not a 4)

Press the +50 button until you have rolled 5000 times. Based on the data, find the following experimental probabilities:

P(2) P(3 or 6) P(odd) P(not a 4)

What can you expect to happen to the experimental probabilities in the long run? In other words, as the number of trials increases, what happens to the experimental probabilities?

Why can there be differences between experimental and theoretical probabilities in general?

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Part 3: Which one do I use?So when do we use theoretical probability or experimental probability? Theoretical probability is always the best choice, when it can be calculated. But sometimes it is not possible to calculate theoretical probabilities because we cannot possible know all of the possible outcomes. In these cases, experimental probability is appropriate. For example, if we wanted to calculate the probability of a student in the class having green as his or her favorite color, we could not use theoretical probability. We would have to collect data on the favorite colors of each member of the class and use experimental probability.

Determine whether theoretical or experimental probability would be appropriate for each of the following. Explain your reasoning:

1. What is the probability of someone tripping on the stairs today between first and second periods?

2. What is the probability of rolling a 3 on a six-sided die, then tossing a coin and getting a head?

3. What is the probability that a student will get 4 of 5 true false questions correct on a quiz?

4. What is the probability that a student is wearing exactly four buttons on his or her clothing today?

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Experimental vs. Theoretical Practice

Name _____________________________________

1) A baseball collector checked 350 cards in case on the shelf and found that 85 of them were damaged. Find the experimental probability of the cards being damaged. Show your work.

2) Jimmy rolls a number cube 30 times. He records that the number 6 was rolled 9 times. According to Jimmy's records, what is the experimental probability of rolling a 6? Show your work.

3) John, Phil, and Mike are going to a bowling match. Suppose the boys randomly sit in the 3 seats next to each other and one of the seats is next to an aisle. What is the probability that John will sit in the seat next to the aisle?

4) In Mrs. Johnson's class there are 12 boys and 16 girls. If Mrs. Johnson draws a name at random, what is the probability that the name will be that of a boy?

5) Antonia has 9 pairs of white socks and 7 pairs of black socks. Without looking, she pulls a black sock from the drawer. What is the probability that the next sock she pulls out will also be black?

6) Lenny tosses a nickel 50 times. It lands heads up 32 times and tails 18 times. What is the experimental probability that the nickel lands tails?

7) A car manufacturer randomly selected 5,000 cars from their production line and found that 85 had some defects. If 100,000 cars are produced by this manufacturer, how many cars can be expected to have defects?

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Unit 6 Review1. Monica came home from school to find a bowl of 4 apples and 4 plums on the table. She decides to have a snack. First she selects one and then puts it back. She then selects another. What is the probability both selections were apples?

2. The Scrabble tiles A, B, E, I, J, K and M are placed face down in the lid of the game and are then mixed up. Two tiles are chosen at random. Find each probability:

a. P(selecting 2 vowels) if no replacement occurs

b. P(selecting 2 vowels) if replacement occurs

c. P(selecting the same letter twice) if no replacement occurs

3. Christine helps her dad do the dishes. There are 5 bowls, 5 glasses, and 6 plates which need to be washed. She accidentally knocks two items off the counter and breaks them. Find each probability:

a. P(breaking 2 plates)

b. P(breaking 2 bowls)

c. P(breaking a bowl and then a glass)

4. Two dice are tossed. Find each probability: a. P(two 3’s)

b. P(no 3’s)

c. P(3 and 4)

d. P(3 and any other number)

5. A jar contains 5 peanut butter cookies, 3 caramel delights, and 7 lemon cookies. If 3 cookies are selected in succession, find the probability of selecting one of each if:

a. no cookies are replaced

b. each cookie is replaced

6. In Jason's homeroom class, there are 12 students who have brown eyes, 5 students who are left-handed, and 3 students who have brown eyes and are left-handed. If there are a total of 27 students in Jason's homeroom class, draw a Venn diagram and find how many of them neither have brown eyes nor are left-handed?7. A coin and a die are tossed. Calculate the probability of getting tails and an even number.

8. If the probability of receiving at least 1 piece of mail on any particular day is 32%, what is the probability of not receiving any mail for 4 days in a row?

9. A card is randomly selected from a standard deck of 52 cards. What is the P(ace or face card)?

10. A card is randomly selected from a standard deck of 52 cards. What is the P(heart or face card)?

11. A pet store contains 36 light green parakeets (15 females and 21 males) and 45 sky blue parakeets (28 females and 17 males). Arrange this information in a two-way table.

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Male Female TotalLight Green ParakeetSky Blue ParakeetTotal

a. You randomly choose one of the parakeets. What is the probability that it is a female or a sky blue parakeet?

b. What is the probability that the randomly chosen parakeet is both green and male?

c. What is the probability that the randomly chosen parakeet is female, given it is green?

12. At Kennedy Middle School, the probability that a student takes Technology and Spanish is 0.07. The probability that a student takes Technology is 0.63. What is the probability that a student takes Spanish given that the student is taking Technology?

13. In New York State, 48% of all teenagers own a skateboard and 39% of all teenagers own a skateboard and roller blades. What is the probability that a teenager owns roller blades given that the teenager owns a skateboard?

14. What is the probability of choosing the ace of clubs from a standard deck of cards given that the card you draw is a black card?15. There are 404 students in the 10th grade. Five of these students will be selected randomly to represent your class on a 5-person bowling team. What is the probability that the team chosen will be you and your 4 best friends?

16. What are all the different ways the letters ABC can be arranged? What is the probability that if you randomly selected one of these arrangements, you would select the one that spells CAB?

17. The weather forecast for Saturday says there is a 40% chance of rain. What are the odds that it will rain on Saturday?

18. What are the odds of drawing an ace at random from a standard deck of cards?

19. From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

20. To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

21. A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

22. What is the probability of spelling the word DOG using the letters A, D, O, G, and P?

23. At the burger shack you can order a burger rare, medium, or well done. It can be plain or have one of these toppings: onions, relish, mayonnaise, cheese, ketchup, or tomato. How many different kinds of burgers can you order?

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24. A restaurant serves five main dishes, three salads, and four desserts. How many different meals could be ordered if each person has a main dish, a salad, and a dessert?

25. A golf club manufacturer makes irons with seven different shaft lengths, three different grips, five different lies, and two different club head materials. How many different combinations are offered?

26. A gymnastics team has 6 American and 4 Romanian girls. Suppose three girls are selected at random from the team. Find the probability that they are all from America.

27. One card is selected at random. Find the probability that the card selected isa. a face card c. a number card less than 7

b. a spade or an odd number card d. a red card or an ace

28. One doughnut is selected from a box of a dozen Dunkin Donuts. In the box, there are 6 glazed, 4 chocolate, and 2 blueberry doughnuts. Half of all the doughnuts have sprinkles. Find the following:

a. P (a chocolate with sprinkles) c. P (a blueberry or a glazed) b. P (a glazed, given that it has sprinkles)

29. Ten jellybeans are placed in a very small bag. There are 4 licorice, 3 cherry, 2 lemon and 1 grape. If three jellybeans are selected, find the probability of selecting:

a. a licorice, then a lemon, then a cherry, with replacement

b. 2 licorice, then a grape, without replacement

c. 3 cherry, without replacement

30. Two regular dice are rolled. Find the following probabilities:a. P(sum of 4 or 8) c. P(first die is a multiple of 3)b. P(first die is prime, second die is odd) d. P(a sum of at most 8)

31. Mr. Williams stops by a class of 30 students and randomly selects 5 students to take out to lunch. Find the following: a . the odds of getting picked to go to lunch

b. the probability of not getting picked

32. The table below shows the results of a survey on favorite ice cream flavors.Vanilla Chocolate Strawberry Total

Male 21 35 12 68Female 17 42 23 82

Total 38 77 35 150a. P(chocolate is the favorite flavor)

b. P(chocolate is selected, given that the person is female)

c. P(strawberry is not selected, given that the person is male)

d. P(a male is selected, given that the flavor is vanilla) 37

33. Lisa has 4 skirts, 8 blouses, and 2 jackets. How many 3-piece outfits can she put together?A) 32 B) 64 C) 14 D) 128

34. A combination lock has 20 numbers on it. How many different 3-digit lock combinations are possible if no digit can be repeated?

A) 2280 B) 6840 C) 1140 D) 38035. A church has 10 bells in its bell tower. Before each church service 3 bells are rung in sequence. No bell is rung more than once. How many sequences are there?

A) 720 B) 604,800 C) 120 D) 1,209,60036. A hamburger shop sells hamburgers with cheese, relish, lettuce, tomato, onion, mustard or ketchup. How many different hamburgers can be concocted using any 5 of the extras?

A) 1260 B) 2520 C) 42 D) 21 37. You randomly select one card from a standard 52-card deck. Then the probability of not selecting a king P(not king) =

A) 1 – P(king) B) 1 + P(king) C) P(king) D) – P(king) 38. The physics department of a college has 7 male professors, 11 female professors, 16 male teaching assistants, and 8 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a teaching assistant or a female.

A) B) C) D)

39. In a class of 50 students, 32 are Democrats, 16 are business majors, and 6 of the business majors are Democrats. If one student is randomly selected form the class, find the probability of choosing a Democrat or a business major.

A) B) C) D)

40. A fair coin is tossed two times in succession. The sample space of equally likely outcomes is (HH, HT, TH, TT). Find the probability of getting the same outcome on each toss.

A) B) C) D) 1

41. You randomly select one card from a standard 52-card deck. Find the probability of selecting an ace or a 9.

A) B) C) D) 10

42. A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of the regions are colored red, two are green, and one is yellow. If the pointer is spun three times, find the probability it will land on green every time.

A) B) C) D)

43. You are dealt one card from a standard 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of getting a picture card the first time and a club the second time.

A) B) C) D)

44. Two dice are rolled. The numbers are multiplied. What is the probability of getting a 12?

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A) B) C) D)

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