LESSON Ready to Go On? Skills Intervention 11-1 Probability · she will not make tuna today? How many choices of sandwich are there? ... Skills Intervention 11-1 Probability LESSON

Copyright © by Holt, Rinehart and Winston. 197 Holt MathematicsAll rights reserved.

Name Date Class

Probability is the measure of how likely an event is to occur. Themore likely an event is to occur, the higher its probability. The lesslikely an event is to occur, the lower its probability. You can findprobability by performing an experiment and recording theoutcome of each trial.

Determining the Likelihood of an EventA basket has apples of various colors: 8 red, 8 green, and2 yellow. All of the apples are the same size and weight.

Which two colors of apples are there Which color apple is there the least of?

more of?

This means that the and This means that the apple is

apples are more likely to be less likely to be chosen.

chosen.

The complement of an event is the probability that you will getanything other than the expected outcome. The sum of theprobability of an outcome and its complement is 1. If the expectedoutcome is x, the probability of its complement is written as P (not x)and is a fraction.

Using ComplementsMarvin’s mom makes him a sandwich of either roast beef, turkey,ham, tuna, or peanut butter every day. What is the probability thatshe will not make tuna today?

How many choices of sandwich are there?

How many choices are not tuna?

What is P (not tuna)?

Weather ApplicationMaribel’s softball team plays every day that it does not rain. It rained13 days in March. What is the probability of randomly choosing aday on the March calendar that Maribel’s team did play?

How many are days in March?

How many days did it rain?

� �

What is P (choosing a day Maribel’s team did play)?

Ready to Go On? Skills InterventionProbability11-1

LESSON

Vocabulary

probabilityevent experimentoutcometrialcomplement


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Ready to Go On? Problem Solving InterventionProbability11-1

LESSON

You can organize what you know about outcomes to help find and compare probabilities.

You roll a number cube with the numbers 1�6. Look at the 4 eventsdescribed below and order them from most likely to least likely.A. Roll an odd number. B. Roll a number divisible by 3.C. Roll a prime number D. Roll a factor of 60.

greater than 5.

Understand the Problem

1. If you roll the number cube, what are the possible outcomes?

2. List the first two prime numbers greater than 5.

3. List the factors of 60.

Make a Plan

4. Which of the outcomes you listed in Exercise 1 are oddnumbers? How does that help you decide how likely Event A is?

5. How might a table like the one below help you solve the problem?

Solve

6. Make a table of the 4 events. In the last row, write impossible,likely, as likely as not, unlikely, or certain.

Check

7. Answer the question in the problem.

Event A B C D

Outcomes that 1, 3, 5Will Work

Probability


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Ready to Go On? Skills InterventionExperimental Probability11-2

LESSON

The ratio of the number of times an event occurs to the totalnumber of trials is called experimental probability.

Sports ApplicationJason has scored 12 touchdowns in the last 5 games. He hasattempted 20 touchdowns total. What is the experimental probabilitythat he will make a touchdown during the next game?

P �

P (touchdown) � � �20

�

The experimental probability for Jason making a touchdown in the

next game is . This can be simplified to �5

�.

Weather ApplicationIn the past 10 years of rain monitoring in Southern California, 5 years have shown less than 17 days of rain.

A. What is the experimental probability that next year will be under17 days of rain?

P � �

� �5�

� �2

�

B. Is it impossible, unlikely, as likely as not, likely, or certain thatduring the next year Southern California will have another yearwith less than 17 days of rain?

How many years has it rained less than 17 days?

How many years has it rained more than 17 days?

Since it has rained less than 17 days the same number of years

that it has rained more than 17 days, it is

that the next year will have less than 17 days.

number of years ��

number of years monitorednumber of times an event occurs��

total number of trials

number of touchdowns made��

total number of

number of times an event occurs��


Vocabulary

experimental probability


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Ready to Go On? Problem Solving InterventionExperimental Probability11-2

LESSON

Sometimes you can find probabilities frominformation displayed in a table.

The table shows the results of a quality control test at a pen factory. Based on these results, what is the probability that a pen chosen at random will be defective?


1. Does the problem ask you to find the probability for a pen froma particular batch? Explain.

Make a Plan

2. Why would it make sense to use the results from all threebatches to determine the probability?

3. Complete with words to show how P(defective) � you will calculate the probability

Solve

4. How many defective pens were found in Batch A? Batch B?Batch C? In all three batches combined?

5. How many pens were tested in all three batches combined?

6. Use the word equation you wrote in Exercise 3 to find the probability that a randomly chosen pen will be defective.

Check

7. Change your answer to a percent and compare it to the separate percents for A, B, and C. Is your answer reasonable?

number of ��

Number PercentBatch in Batch Defective

A 1,000 1.5%

B 1,500 2.0%

C 2,500 1.0%


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Ready to Go On? Skills InterventionMake a List to Find Sample Spaces11-3

LESSON

The sample space is made up of all the possible outcomes of anexperiment. The Fundamental Counting Principle states thatyou can find the total number of outcomes for 2 or moreexperiments by multiplying the number of outcomes for eachexperiment.

Making a List to Find a Sample SpaceBarry plays a dart game at the local carnival. For $1.00 he gets twoattempts to hit a red star 15 feet away. What are all the possibleoutcomes? How large is the sample space?

1. Understand the ProblemRewrite the question as a statement.

• Find all of the of hitting a red

star and determine the .

List the important information.

• There are attempts.

• Each dart can either land or .

2. Make a Plan

You can make a to track all the possibleoutcomes.

3. Solve

Let H = Hits and M = Misses.

The possible outcomes are .

There are possible outcomes.

4. Check

Is each possible outcome that is recorded in the list different?

Vocabulary

sample spaceFundamental

CountingPrinciple

Attempt 1 Attempt 2

H

M

M

M M


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Ready to Go On? Problem Solving InterventionMake a List to Find Sample Spaces11-3

LESSON

Organizing with a table can help you list the outcomes in a sample space.

You roll a number cube labeled 1�6 twice and add the twonumbers you get. What are all the possible outcomes in the sample space? How large is the sample space?


1. Is rolling a 1 and a 2 an outcome? List another outcome.

2. Is rolling 1 and 2 the same outcome as rolling 2 and 1 becauseboth have sums of 3? Explain.

Make a Plan

3. Why might a table be helpful?

Solve

4. Complete the table to show each outcome.

5. How many outcomes are in the sample space?

Check

6. Make sure each outcome is different and none are left out.

Solve

7. How many outcomes would there be if you used two 8-sided number cubes numbered 1–8 instead of two 6-sided number cube?

8. How many outcomes would there be if you used three 4-sided number cubes, each numbered 1–4?

1 2 3 4 5 6

1 1�1 1�2 1�3

2

3

4

5

6


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Ready to Go On? Skills InterventionTheoretical Probability11-4

LESSON

The theoretical probability of an event is the ratio of the numberof favorable outcomes to the number of possible outcomes.

Finding Theoretical ProbabilityAll of the letters of the word MISSISSIPPI are placed in a bag.Find the probability of drawing an S. Write your answer as afraction, decimal, and a percent. Round to the nearest hundredth.

P �

P (choosing S) �

� �4�

Fraction Decimal Percent

�141� � 4 � 11 � � • 100 � %

School ApplicationMrs. Hart has 12 plastic balls with the numbers 1–12 written onthem. She will choose 6 of them to see who won the raffle.

A. What is the probability that she will pick an even number? Writeyour answer as a fraction, decimal, and a percent.

P(even number) �

� �6�


�162� � � 1 � 2 � � • 100 � %

B. The first five balls Mrs. Hart chose were 2, 4, 5, 7, and 9. Basedon these results, what is the probability that the sixth ball will beeven?

P(even number) �

� �7

�

� 0.

number of even numbers left��

total number of balls left

number of even numbers��

total number of balls

number of S ’s��total number of letters

number of favorable outcomes��number of possible outcomes

Vocabulary

theoretical probability


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11-1 ProbabilityFor each event, determine whether it is impossible, unlikely, aslikely as not, likely, or certain.

1. Pablo draws an ace from a face-down full deck of cards that has been well shuffled.

2. Araceli tosses two number cubes and rolls 13.

3. Myrna draws a red card from a face-down deck of cards with no jokers that has been well shuffled.

4. Adelle is tossing a softball up in the air in an open field and it comes back down.

5. Chan is tossing a balanced coin and it lands heads up.

6. Given that she has just hit a ground ball into the outfield during a softball game, Julia reaches first base safely without being put out.

7. Wolfgang gets straight A’s in all his subjects without ever opening a book.

8. Woodrow plays a complete game of checkers and the result is a win, a loss, or a draw.

11-2 Experimental Probability

9. While riding along a bike path, Adam counts 21 bicycles with thin tires and 48 bicycles with thick tires. Give the experimental probability that the next bicycle Adam sees will have thick tires.

10. Felicia tosses a pair of coins 30 times. She gets double tails 8 times. What is the experimental probability that the next toss will result in double tails?

11. Butch is looking out the kitchen window. He finds that 17 out of the first 33 birds he sees are sparrows. What is the experimental probability that the next bird he sees will not be a sparrow?

12. In a corner parking lot, 6 of the cars are red and 15 of the cars are not red. Give the experimental probability that the next car that enters the parking lot will be red.

Ready to Go On? Quiz11ASECTION


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11-3 Make a List to Find Sample Spaces

13. In a restaurant, Darryl has a choice of 3 different salads and 5 different salad dressings. How many different combinations of salad and dressing does Darryl have to choose from?

14. Wilma rolls two number cubes, one yellow and the other green. If the color matters, how many different outcomes are possible?

15. Teena puts three strips of paper in a hat, one red, one yellow, and one blue. First, she draws one of the strips of paper from the hat randomly. Then she flips a coin. How many different outcomes are possible?

Name them.

16. A pizzeria offers 8 different kinds of pizza and 4 different kinds of soda. How many different combinations of pizza and soda do you have to choose from if you order one of each?

17. There are 11 boys and 8 girls at a party. How many different combinations of one boy and one girl are possible in order to form a dance couple?

11-4 Theoretical ProbabilityA spinner with 8 equal sections numbered from 1 through 8 isbeing spun. Find the probability of each event. Write your answeras a fraction in simplest form, as a decimal, and as a percent.

18. P(3) 19. P(odd number)

20. P(10) 21. P(prime number)

22. P(composite number) 23. P(number � 0)

24. P(perfect square) 25. P(2 � number � 8)

Ready to Go On? Quiz continued

11ASECTION

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The exercises on this page are all to be done by rolling a pairof number cubes, numbered from 1 through 6.

1. What is the theoretical probability of rolling the same number on both cubes?

2. Roll both cubes. Record 1 if the two numbers are the same. Record 0 if they are different.

3. Roll the two cubes again. As before, record 1 for “the same” or 0 for “different.”

4. Roll the cubes 18 more times. Each time record 1 for “the same” or 0 for “different.”

5. Based on your 20 rolls of the dice, what is the experimental probability that the two numbers rolled will be the same?

6. Is this experimental probability close to your theoretical probability in Exercise 1?

7. Do another test, rolling the two cubes 20 times. Record 1 or 0 each time.

8. What is your experimental probability this time?

9. Is this number close to your first experimental probability?

10. Check with some of your classmates. Record the experimental probabilities they got.

11. Are the experimental probabilities your classmates got close to the ones you recorded?

Ready to Go On? EnrichmentRolling Number Cubes11A

SECTION

MSM07C2_RTGO_ch11_197-215_B 6/17/06 3:32 PM Page 206 (Black plate)


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Events are dependent if one event has an effect on the secondevent’s probability. Events are independent if one event has noeffect on the second event’s occurrence.

Determining Whether Events are Independent or DependentDecide if the event is independent or dependent.

Ronny chooses a king of hearts from a deck of cards and then rollsa 5 on a fair number cube.

Did choosing a card affect Ronny’s rolling the number cube?

Is this set of events independent or dependent?

Finding the Probability of Independent EventsPeter wakes up late half the time and misses the bus. As a result hehas to take a taxi to work. His best friend carpools with 4 otherfriends, each having to drive their car to work one day out of theweek. What is the probability that Peter is using a taxi and his bestfriend is driving the carpool?

P(taxi and best friend driving) � P(taxi) • P(best friend driving)

� �2

� • �1

� There are options for Peter.

There are different people.

�

The probability is .

Finding the Probability of Dependent EventsSuzie goes on a picnic and takes 5 apples, 4 oranges, and6 bananas. She eats a single fruit at a time. What is the probabilitythat Suzie eats two apples in a row?

P(A and B) � P(A) • P(B)

What is the first event, A? Suzie eating the apple.

What is the second event, B? Suzie eating the apple.

What is the probability that Suzie eats an apple? �15

� � �13

�

What is the probability that Suzie eats a second apple? �4�

P(Suzie eats two apples in a row) � �155� • �

144� � � �

21�

Ready to Go On? Skills InterventionProbability of Independent and Dependent Events11-5

LESSON

Vocabulary

dependent eventsindependent

events


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Ready to Go On? Problem Solving InterventionProbability of Independent and Dependent Events11-5

LESSON

In contest A, you randomly pick 5 cards one at a time from a bag. You must pick them in the exact order 1�5 to win. In contest B, you spin the spinner 5 times. If you land on gray each time, you win. Which contest is easier to win?


1. In contest A, what number must you get on your first pick in order to have any chance of winning?

2. In contest B, what must you get on your first spin in order to have any chance of winning?

3. What probabilities will you compare?

Make a Plan

4. Does picking a card change the cards that are left? Does thatmean the events in contest A are independent or dependent?

5. Does spinning the spinner change the chances on the nextspin? Are the events in contest B independent or dependent?

Solve

6. Complete to find P(A), the probability of winning contest A.

P(A) � P(1) • P(2 after 1) • P(3 after 1 � 2) • P(4 after 1 � 3) • P(5 after 1 � 4)

� ��1

�� 14

�� 1

��

7. Complete to find P(B), the probability of winning contest B.

P(B) � P(gray) • P(gray) • P(gray) • P(gray) • P(gray)

� ��1

��

Check


Contest B

2

134

5

Contest A


Name Date Class

Ready to Go On? Skills InterventionCombinations11-6

LESSON

A grouping of objects or events in which order does not matter iscalled a combination.

Using a Table to Find CombinationsDavid rents bikes, skates, skateboards, and surfboards at thebeach. For people who choose to rent two different items, hediscounts the price 25%. How many combinations of two-itemrentals are there?

List all of the possible two-item groupings.

Since order does not matter, cross out repeated pairs.

Which possible two-item groupings remain?

How many two-item groupings are left?

To get a 25% discount, customers can rent different two-item combinations.

Vocabulary

combination

Skates Skateboards Surfboards BikesSkateboards Skates Skates SkatesSkates Skateboards Surfboards BikesSurfboards Surfboards Skateboards SkateboardsSkates Skateboards Surfboards BikesBikes Bikes Bikes Surfboards

SkatesSkateboardsSkates SkateboardsSurfboards SurfboardsSkates Skateboards SurfboardsBikes Bikes Bikes

Skates Skateboards Surfboards BikesSkateboards Skates Skates Skates

Skates Skateboards Surfboards BikesSurfboards Surfboards Skateboards Skateboards

Skates Skateboards Surfboards BikesBikes Bikes Bikes Surfboards


Name Date Class

Sometimes you can make a problem much simpler just by looking atit a different way.

You and a friend each have a set of 26 alphabet cards labeled A�Z.You pick 5 cards at random. You win if you pick the 5 vowels. Yourfriend picks 21 cards at random and wins if she picks all 21consonants. Who has a better chance of winning?


1. If you pick the 5 vowels, does it matter in what order you pick them?

2. Does the problem ask you to find the probabilities? What does it ask?

Make a Plan

3. Why might you try logical reasoning to solve the problem?

Solve

4. Suppose it turned out that the number of possible combinations of 5 cards is thesame as the number of possible combinations of 21 cards. How would that help you solve the problem?

5. When you pick a combination of 5 cards, what’s left in the pack?

6. Why does your answer to part 5 mean that the number of 5-card combinationsequals the number of 21-card combinations?

Check


Ready to Go On? Problem Solving InterventionCombinations 11-6

LESSON


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Ready to Go On? Skills InterventionPermutations 11-7

LESSON

A permutation is an arrangement of objects or events whereorder does matter.

Using the Fundamental Counting Principle to Find theNumber of PermutationsJean, Sally, and Kate have all signed up to run in the annualmarathon. They are favored to come in first, second, and thirdplace.

A. How many different permutations are there?

There are choices for first place.

There are remaining choices for second place.

There is choice for third place.

3 • • �

There are different permutations.

B. List the different permutations. Complete the table.

C. What is the probability that Kate will come in first place?

P(Kate comes in first) �

� �6

�

�

The probability that Kate will come in first place is .

Number of Permutations with Kate in First��

Total Number of Permutations

Vocabulary

permutation

Permutations First Second Third

1 Jean Sally Kate2 Jean Kate Sally3 Sally Jean Kate4 Sally Kate Jean5 Kate Sally Jean6 Kate Jean Sally


Name Date Class

To solve some probability problems, you must decide if ordermatters.

You have nine cards numbered 1�9. You randomly draw two cardsone at a time without replacing them. You form a 2-digit number byusing the first card as the tens digit and the second as the ones digit.What is the probability that the number you form is divisible by 5?


1. Why can’t you form the number 8?

2. If you get 7 on the first draw, why can’t you get 7 on the next draw?

3. Would it be a favorable outcome if you drew 3 then 5? If youdrew 5 then 3? List another favorable outcome.

Make a Plan

4. How can you use outcomes to calculate the probability of an event?

Solve

5. How many different 2-digit numbers can be formed? Explain.

6. Which 2-digit numbers divisible by 5 cannot be formed? Explain.

7. How many favorable outcomes are there?

8. What is the probability that the number formed is divisible by 5?

Check

9. Check your list of favorable outcomes for numbers that cannot be formed.

Ready to Go On? Problem Solving InterventionPermutations 11-7

LESSON


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11-5 Probability of Independent and Dependent EventsDecide whether each set of events is independent ordependent. Explain your answer.

1. Alicia flips a dime and a penny. The dime comes up heads andthe penny comes up tails.

2. A box has ten names placed in it. First, Pedro draws a name randomly fromthe box. Then, Kazuko randomly draws one of the remaining names.

3. Graciela selects a card randomly from a full deck of playingcards and looks at it before replacing it. Then, Richard carefullyshuffles the deck and selects a card.

4. Abdul picks one of the ten digits randomly using his calculator.Then, again using his calculator, he picks one of the nineremaining digits.

A hat has the names of 5 boys and 7 girls.

5. What is the probability of randomly selecting a girl from the names in the hat, replacing her name, and then randomly selecting a boy?

6. What is the probability of randomly selecting a boy from the names in the hat and then randomly selecting a girl without replacing the first name?

All but the 13 clubs have been removed from a deck of shuffledplaying cards.

7. What is the probability of drawing a face card and then another face card if there is no replacement?

8. What is the probability of drawing a face card and then another face card if there is replacement?

Ready to Go On? Quiz11BSECTION


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11-6 Combinations

9. An ice-cream shop has 10 different flavors. Frieda wants to buy an ice-cream cone with 2 scoops and different flavors. How many combinations of two different flavors are available?

10. Bill and 7 of his friends want to divide up into two teams to play some touch football. There will be 4 players on each team. In how many different ways can the members of the two teams be picked?

11. Mimi is buying board games to give her friends as gifts. There are 7 different games at the store. How many combinations of 3 different games are available?

12. Charles finds 9 different kinds of soup in the grocery store. He wants to buy 5 of these. How many combinations of 5 different kinds of soup are available?

13. Louisa and 5 of her friends want to divide up into two groups for a school project. There will be exactly 3 people in each group. In how many different ways can the members of the two groups be chosen?

11-7 Permutations

14. Bill’s touch football team has 4 players on it. In how many different ways can these 4 players be arranged in 4 different playing positions?

15. In how many different orders can Luann arrange 6 books on one of her bookshelves?

16. The school chess club has 9 members. Only 5 members at a time can occupy positions on the chess club team, which has a top board, second board, third board, and so on. In how many different ways can these 5 positions be filled?

17. Another school’s chess club has exactly 5 members. In how many different ways can the same 5 positions on the club chess team be filled?

18. The city is considering 8 students for 6 summer jobs. The 6 jobs are all different. How many ways are there of filling the 6 jobs?


11BSECTION


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1. All but the 13 diamonds have been removed from a deck of playing cards. What is the probability of randomly drawing a face card (king, queen, or jack) from the deck?

2. Represent this on the array by shading the three horizontal rows K, Q, and J yellow. What fraction of the rows is shaded?

3. If the drawn card is replaced, what is the probability of randomly drawing a face card on a second draw?

4. Represent this by shading the three vertical columns K, Q, and J red. What fraction of the columns is shaded?

5. What is the probability of drawing a face card, replacing it, and then drawing a face card again?

6. What fraction of the small squares in the array is shaded both yellow and red?

7. Now compare your answers to questions 5 and 6. What do you notice about them?

8. If two cards are drawn without replacement from our deck of 13 cards, they cannot both be the same card. Draw an Xthrough each small square that represents two aces, two 2’s, two 3’s, and so on. How many squares have you crossed out?

9. Two cards are drawn without replacement. What is the probability that both are face cards?

10. Excluding all the squares that are crossed out, what fraction of the small squares are shaded both yellow and red?

11. What do you notice about your last two answers?

Ready to Go On? EnrichmentProbability and Arrays11B

SECTION

A 2 3 4 5 6 7 8 9 10 J Q K

A

2

3

4

5

6

7

8

9

10

J

Q

K



Probability is the measure of how likely an event is to occur. Themore likely an event is to occur, the higher its probability. The lesslikely an event is to occur, the lower its probability. You can findprobability by performing an experiment and recording theoutcome of each trial.

Determining the Likelihood of an EventA basket has apples of various colors: 8 red, 8 green, and2 yellow. All of the apples are the same size and weight.

Which two colors of apples are there Which color apple is there the least of?

more of?

This means that the and This means that the apple is

apples are more likely to be less likely to be chosen.

chosen.

The complement of an event is the probability that you will getanything other than the expected outcome. The sum of theprobability of an outcome and its complement is 1. If the expectedoutcome is x, the probability of its complement is written as P (not x)and is a fraction.

Using ComplementsMarvin’s mom makes him a sandwich of either roast beef, turkey,ham, tuna, or peanut butter every day. What is the probability thatshe will not make tuna today?

How many choices of sandwich are there?

How many choices are not tuna?

What is P (not tuna)?

Weather ApplicationMaribel’s softball team plays every day that it does not rain. It rained13 days in March. What is the probability of randomly choosing aday on the March calendar that Maribel’s team did play?

How many are days in March?

How many days did it rain?

� �

What is P (choosing a day Maribel’s team did play)? �1381�

31181313

31

�45

�

45

greenyellowred

yellowred and green

Ready to Go On? Skills InterventionProbability11-1

LESSON

Vocabulary

probabilityevent experimentoutcometrialcomplement


Ready to Go On? Problem Solving InterventionProbability11-1

LESSON

You can organize what you know about outcomes to help find and compare probabilities.

You roll a number cube with the numbers 1�6. Look at the 4 eventsdescribed below and order them from most likely to least likely.A. Roll an odd number. B. Roll a number divisible by 3.C. Roll a prime number D. Roll a factor of 60.

greater than 5.


1. If you roll the number cube, what are the possible outcomes?

2. List the first two prime numbers greater than 5.

3. List the factors of 60.

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Make a Plan

4. Which of the outcomes you listed in Exercise 1 are oddnumbers? How does that help you decide how likely Event A is?

1, 3, and 5; Sample response: I can compare how many odd numbers

there are with how many even numbers there are.5. How might a table like the one below help you solve the problem?

Sample answer: It can help me keep track of the outcomes and

probabilities.

Solve

6. Make a table of the 4 events. In the last row, write impossible,likely, as likely as not, unlikely, or certain.

Check

7. Answer the question in the problem. D, A, B, C

7, 111, 2, 3, 4, 5, 6

Event A B C D

Outcomes that 1, 3, 5 3, 6 none 1, 2, 3,Will Work 4, 5, 6Probability as likely unlikely impossible certain

as not


Ready to Go On? Skills InterventionExperimental Probability11-2

LESSON

The ratio of the number of times an event occurs to the totalnumber of trials is called experimental probability.

Sports ApplicationJason has scored 12 touchdowns in the last 5 games. He hasattempted 20 touchdowns total. What is the experimental probabilitythat he will make a touchdown during the next game?

P �

P (touchdown) � � �1220�

The experimental probability for Jason making a touchdown in the

next game is . This can be simplified to �35

�.

Weather ApplicationIn the past 10 years of rain monitoring in Southern California, 5 years have shown less than 17 days of rain.

A. What is the experimental probability that next year will be under17 days of rain?

P � �

� �150�

� �12

�

B. Is it impossible, unlikely, as likely as not, likely, or certain thatduring the next year Southern California will have another yearwith less than 17 days of rain?

How many years has it rained less than 17 days?

How many years has it rained more than 17 days?

Since it has rained less than 17 days the same number of years

that it has rained more than 17 days, it is

that the next year will have less than 17 days.

as likely as not

55

number of years it rained under 17 days��

number of years monitorednumber of times an event occurs��


�1220�

number of touchdowns made��

total number of attempts

number of times an event occurs��


Vocabulary

experimental probability


Ready to Go On? Problem Solving InterventionExperimental Probability11-2

LESSON

Sometimes you can find probabilities frominformation displayed in a table.

The table shows the results of a quality control test at a pen factory. Based on these results, what is the probability that a pen chosen at random will be defective?


1. Does the problem ask you to find the probability for a pen froma particular batch? Explain.

no; It asks about a “pen chosen at random.”

Make a Plan

2. Why would it make sense to use the results from all threebatches to determine the probability?

Sample answer: I’m finding the probability of a randomly chosen pen, so

I do not know which batch it comes from.3. Complete with words to show how P(defective) �

you will calculate the probability

Solve

4. How many defective pens were found in Batch A? Batch B?Batch C? In all three batches combined?

15; 30; 25; 70

5. How many pens were tested in all three batches combined?

6. Use the word equation you wrote in Exercise 3 to find the probability that a randomly chosen pen will be defective.

P(defective) � �5,

70000� � �

5700�

Check

7. Change your answer to a percent and compare it to the separate percents for A, B, and C. Is your answer reasonable?

Sample response: Yes, �5700� � 1.4%, which is close to the

percents shown.

5,000

number of defective pens��

total number of pens

Number PercentBatch in Batch Defective

A 1,000 1.5%

B 1,500 2.0%

C 2,500 1.0%



Ready to Go On? Skills InterventionMake a List to Find Sample Spaces11-3

LESSON

The sample space is made up of all the possible outcomes of anexperiment. The Fundamental Counting Principle states thatyou can find the total number of outcomes for 2 or moreexperiments by multiplying the number of outcomes for eachexperiment.

Making a List to Find a Sample SpaceBarry plays a dart game at the local carnival. For $1.00 he gets twoattempts to hit a red star 15 feet away. What are all the possibleoutcomes? How large is the sample space?

1. Understand the ProblemRewrite the question as a statement.

• Find all of the of hitting a red

star and determine the .

List the important information.

• There are attempts.

• Each dart can either land or .

2. Make a Plan

You can make a to track all the possibleoutcomes.

3. Solve

Let H = Hits and M = Misses.

The possible outcomes are .

There are possible outcomes.

4. Check

Is each possible outcome that is recorded in the list different? yes

4HH, HM, MH, MM

list

not on the staron the startwo

sample spacepossible outcomes

Vocabulary

sample spaceFundamental

CountingPrinciple

Attempt 1 Attempt 2

H HH M

M HM M


Ready to Go On? Problem Solving InterventionMake a List to Find Sample Spaces11-3

LESSON

Organizing with a table can help you list the outcomes in a sample space.

You roll a number cube labeled 1�6 twice and add the twonumbers you get. What are all the possible outcomes in the sample space? How large is the sample space?


1. Is rolling a 1 and a 2 an outcome? List another outcome.

yes; Sample answer: rolling two 4’s2. Is rolling 1 and 2 the same outcome as rolling 2 and 1 because

both have sums of 3? Explain.

The outcomes are different because the sequence of the rolled

numbers is different. The sums of the rolled numbers are the same.

Make a Plan

3. Why might a table be helpful?

Sample answer: It makes it easier to show all the possible pairs of

numbers that could be rolled and the sum of each pair.

Solve

4. Complete the table to show each outcome.

5. How many outcomes are in the sample space?

36

Check

6. Make sure each outcome is different and none are left out.

Solve

7. How many outcomes would there be if you used two 8-sided number cubes numbered 1–8 instead of two 6-sided number cube?

8. How many outcomes would there be if you used three 4-sided number cubes, each numbered 1–4? 64

64

1 2 3 4 5 6

1 1�1 1�2 1�3 1�4 1�5 1�62 2�1 2�2 2�3 2�4 2�5 2�63 3�1 3�2 3�3 3�4 3�5 3�64 4�1 4�2 4�3 4�4 4�5 4�65 5�1 5�2 5�3 5�4 5�5 5�66 6�1 6�2 6�3 6�4 6�5 6�6


Ready to Go On? Skills InterventionTheoretical Probability11-4

LESSON

The theoretical probability of an event is the ratio of the numberof favorable outcomes to the number of possible outcomes.

Finding Theoretical ProbabilityAll of the letters of the word MISSISSIPPI are placed in a bag.Find the probability of drawing an S. Write your answer as afraction, decimal, and a percent. Round to the nearest hundredth.

P �

P (choosing S) �

� �141�


�141� � 4 � 11 � � • 100 � %

School ApplicationMrs. Hart has 12 plastic balls with the numbers 1–12 written onthem. She will choose 6 of them to see who won the raffle.

A. What is the probability that she will pick an even number? Writeyour answer as a fraction, decimal, and a percent.

P(even number) �

� �162�


�162� � � 1 � 2 � � • 100 � %

B. The first five balls Mrs. Hart chose were 2, 4, 5, 7, and 9. Basedon these results, what is the probability that the sixth ball will beeven?

P(even number) �

� �47

�

� 0. 57

number of even numbers left��

total number of balls left

500.500.50�12

�

number of even numbers��

total number of balls

36.360.36�0.36�

number of S ’s��total number of letters

number of favorable outcomes��number of possible outcomes

Vocabulary

theoretical probability


11-1 ProbabilityFor each event, determine whether it is impossible, unlikely, aslikely as not, likely, or certain.

1. Pablo draws an ace from a face-down full deck of cards that has been well shuffled.

2. Araceli tosses two number cubes and rolls 13.

3. Myrna draws a red card from a face-down deck of cards with no jokers that has been well shuffled.

4. Adelle is tossing a softball up in the air in an open field and it comes back down.

5. Chan is tossing a balanced coin and it lands heads up.

6. Given that she has just hit a ground ball into the outfield during a softball game, Julia reaches first base safely without being put out.

7. Wolfgang gets straight A’s in all his subjects without ever opening a book.

8. Woodrow plays a complete game of checkers and the result is a win, a loss, or a draw.

11-2 Experimental Probability

9. While riding along a bike path, Adam counts 21 bicycles with thin tires and 48 bicycles with thick tires. Give the experimental probability that the next bicycle Adam sees will have thick tires.

10. Felicia tosses a pair of coins 30 times. She gets double tails 8 times. What is the experimental probability that the next toss will result in double tails?

11. Butch is looking out the kitchen window. He finds that 17 out of the first 33 birds he sees are sparrows. What is the experimental probability that the next bird he sees will not be a sparrow?

12. In a corner parking lot, 6 of the cars are red and 15 of the cars are not red. Give the experimental probability that the next car that enters the parking lot will be red.

�27

�

�1363�

�145�

�1263�

certain

unlikely

likely

as likely as notcertain

as likely as not

impossibleunlikely

Ready to Go On? Quiz11ASECTION



11-3 Make a List to Find Sample Spaces

13. In a restaurant, Darryl has a choice of 3 different salads and 5 different salad dressings. How many different combinations of salad and dressing does Darryl have to choose from?

14. Wilma rolls two number cubes, one yellow and the other green. If the color matters, how many different outcomes are possible?

15. Teena puts three strips of paper in a hat, one red, one yellow, and one blue. First, she draws one of the strips of paper from the hat randomly. Then she flips a coin. How many different outcomes are possible?

Name them.

red-heads, red-tails, blue-heads, blue-tails, yellow-heads, yellow-tails16. A pizzeria offers 8 different kinds of pizza and 4 different

kinds of soda. How many different combinations of pizza and soda do you have to choose from if you order one of each?

17. There are 11 boys and 8 girls at a party. How many different combinations of one boy and one girl are possible in order to form a dance couple?

11-4 Theoretical ProbabilityA spinner with 8 equal sections numbered from 1 through 8 isbeing spun. Find the probability of each event. Write your answeras a fraction in simplest form, as a decimal, and as a percent.

18. P(3) 19. P(odd number)

20. P(10) 21. P(prime number)

22. P(composite number) 23. P(number � 0)

24. P(perfect square) 25. P(2 � number � 8)

�58

�, 0.625, 62.5%�14

�, 0.25, 25%

1, 1, 100%�38

�, 0.375, 37.5%

�12

�, 0.5, 50%0, 0, 0%

�12

�, 0.5, 50%�18

�, 0.125, 12.5%

88

32

6

36

15


11ASECTION


The exercises on this page are all to be done by rolling a pairof number cubes, numbered from 1 through 6.

1. What is the theoretical probability of rolling the same number on both cubes?

2. Roll both cubes. Record 1 if the two numbers are the same. Record 0 if they are different.

3. Roll the two cubes again. As before, record 1 for “the same” or 0 for “different.”

4. Roll the cubes 18 more times. Each time record 1 for “the same” or 0 for “different.”

5. Based on your 20 rolls of the dice, what is the experimental probability that the two numbers rolled will be the same?

6. Is this experimental probability close to your theoretical probability in Exercise 1?

7. Do another test, rolling the two cubes 20 times. Record 1 or 0 each time.

8. What is your experimental probability this time?

9. Is this number close to your first experimental probability?

10. Check with some of your classmates. Record the experimental probabilities they got.

11. Are the experimental probabilities your classmates got close to the ones you recorded? Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

�16

�

Ready to Go On? EnrichmentRolling Number Cubes11A

SECTION


Events are dependent if one event has an effect on the secondevent’s probability. Events are independent if one event has noeffect on the second event’s occurrence.

Determining Whether Events are Independent or DependentDecide if the event is independent or dependent.

Ronny chooses a king of hearts from a deck of cards and then rollsa 5 on a fair number cube.

Did choosing a card affect Ronny’s rolling the number cube?

Is this set of events independent or dependent?

Finding the Probability of Independent EventsPeter wakes up late half the time and misses the bus. As a result hehas to take a taxi to work. His best friend carpools with 4 otherfriends, each having to drive their car to work one day out of theweek. What is the probability that Peter is using a taxi and his bestfriend is driving the carpool?

P(taxi and best friend driving) � P(taxi) • P(best friend driving)

� �12

� • �15

� There are options for Peter.

There are different people.

�

The probability is .

Finding the Probability of Dependent EventsSuzie goes on a picnic and takes 5 apples, 4 oranges, and6 bananas. She eats a single fruit at a time. What is the probabilitythat Suzie eats two apples in a row?

P(A and B) � P(A) • P(B)

What is the first event, A? Suzie eating the apple.

What is the second event, B? Suzie eating the apple.

What is the probability that Suzie eats an apple? �155� � �

13

�

What is the probability that Suzie eats a second apple? �144�

P(Suzie eats two apples in a row) � �155� • �

144� � � �

221�

�22100

�

secondfirst

�110�

�110�

52

independentNo

Ready to Go On? Skills InterventionProbability of Independent and Dependent Events11-5

LESSON

Vocabulary

dependent eventsindependent

events


Ready to Go On? Problem Solving InterventionProbability of Independent and Dependent Events11-5

LESSON

In contest A, you randomly pick 5 cards one at a time from a bag. You must pick them in the exact order 1�5 to win. In contest B, you spin the spinner 5 times. If you land on gray each time, you win. Which contest is easier to win?


1. In contest A, what number must you get on your first pick in order to have any chance of winning?

2. In contest B, what must you get on your first spin in order to have any chance of winning?

3. What probabilities will you compare?

P of picking the cards in order and P of spinning gray 5 times.

Make a Plan

4. Does picking a card change the cards that are left? Does thatmean the events in contest A are independent or dependent?

yes; dependent 5. Does spinning the spinner change the chances on the next

spin? Are the events in contest B independent or dependent?

no; independent

Solve

6. Complete to find P(A), the probability of winning contest A.

P(A) � P(1) • P(2 after 1) • P(3 after 1 � 2) • P(4 after 1 � 3) • P(5 after 1 � 4)

� ��15

�� 14

�� 13

�� 12

�� 11

�� 1

120�

7. Complete to find P(B), the probability of winning contest B.

P(B) � P(gray) • P(gray) • P(gray) • P(gray) • P(gray)

� ��13

�� 13

�� 13

�� 13

�� 13

�� 2

143�

Check


Contest A is easier to win.

gray

1

Contest B

2

134

5

Contest A

MSM07C2_RTGO_AK_234-292 6/17/06 3:57 PM Page 285



Ready to Go On? Skills InterventionCombinations11-6

LESSON

A grouping of objects or events in which order does not matter iscalled a combination.

Using a Table to Find CombinationsDavid rents bikes, skates, skateboards, and surfboards at thebeach. For people who choose to rent two different items, hediscounts the price 25%. How many combinations of two-itemrentals are there?

List all of the possible two-item groupings.

Since order does not matter, cross out repeated pairs.

Which possible two-item groupings remain?

How many two-item groupings are left?

To get a 25% discount, customers can rent different two-item combinations.

66

Vocabulary

combination

Skates Skateboards Surfboards BikesSkateboards Skates Skates SkatesSkates Skateboards Surfboards BikesSurfboards Surfboards Skateboards SkateboardsSkates Skateboards Surfboards BikesBikes Bikes Bikes Surfboards

SkatesSkateboardsSkates SkateboardsSurfboards SurfboardsSkates Skateboards SurfboardsBikes Bikes Bikes

Skates Skateboards Surfboards BikesSkateboards Skates Skates Skates

Skates Skateboards Surfboards BikesSurfboards Surfboards Skateboards Skateboards

Skates Skateboards Surfboards BikesBikes Bikes Bikes Surfboards


Sometimes you can make a problem much simpler just by looking atit a different way.

You and a friend each have a set of 26 alphabet cards labeled A�Z.You pick 5 cards at random. You win if you pick the 5 vowels. Yourfriend picks 21 cards at random and wins if she picks all 21consonants. Who has a better chance of winning?


1. If you pick the 5 vowels, does it matter in what order you pick them?

2. Does the problem ask you to find the probabilities? What does it ask?

no; Which probability is greater, picking the 5 vowels or picking

the 21 consonants?

Make a Plan

3. Why might you try logical reasoning to solve the problem?

Sample response: The calculation seems very complicated, so it pays

to try looking for a shortcut.

Solve

4. Suppose it turned out that the number of possible combinations of 5 cards is thesame as the number of possible combinations of 21 cards. How would that help you solve the problem?

The probabilities would be equal. So, my friend and I would have

equal chances of winning.

5. When you pick a combination of 5 cards, what’s left in the pack?

6. Why does your answer to part 5 mean that the number of 5-card combinationsequals the number of 21-card combinations?

For each 5 letters you pick, there are 21 letters left, which is the same

thing as a 21-letter combination.

Check


There is an equal chance of winning.

21 cards

no

Ready to Go On? Problem Solving InterventionCombinations 11-6

LESSON


Ready to Go On? Skills InterventionPermutations 11-7

LESSON

A permutation is an arrangement of objects or events whereorder does matter.

Using the Fundamental Counting Principle to Find theNumber of PermutationsJean, Sally, and Kate have all signed up to run in the annualmarathon. They are favored to come in first, second, and thirdplace.

A. How many different permutations are there?

There are choices for first place.

There are remaining choices for second place.

There is choice for third place.

3 • • �

There are different permutations.

B. List the different permutations. Complete the table.

C. What is the probability that Kate will come in first place?

P(Kate comes in first) �

� �26

�

�

The probability that Kate will come in first place is .�13

�

�13

�

Number of Permutations with Kate in First��

Total Number of Permutations

6612

onetwo

three

Vocabulary

permutation

Permutations First Second Third

1 Jean Sally Kate2 Jean Kate Sally3 Sally Jean Kate4 Sally Kate Jean5 Kate Sally Jean6 Kate Jean Sally


To solve some probability problems, you must decide if ordermatters.

You have nine cards numbered 1�9. You randomly draw two cardsone at a time without replacing them. You form a 2-digit number byusing the first card as the tens digit and the second as the ones digit.What is the probability that the number you form is divisible by 5?


1. Why can’t you form the number 8?

Only 2-digit numbers may be formed.2. If you get 7 on the first draw, why can’t you get 7 on the next draw?

A card is not put back after it’s drawn, and there is only one card with

7 on it.3. Would it be a favorable outcome if you drew 3 then 5? If you

drew 5 then 3? List another favorable outcome.

yes; no; Sample answer: 4 then 5.

Make a Plan

4. How can you use outcomes to calculate the probability of an event?

Solve

5. How many different 2-digit numbers can be formed? Explain.

72; there are 9 possible digits for the first draw and 8 for the second

draw, so the total is 9 • 8, which is 72.6. Which 2-digit numbers divisible by 5 cannot be formed? Explain.

multiples of 10 (there is no 0 card) and 55 (5 can’t be drawn twice)7. How many favorable outcomes are there?

8. What is the probability that the number formed is divisible by 5?

Check

9. Check your list of favorable outcomes for numbers that cannot be formed.

�782�, or �

19

�

8

number of favorable outcomes ��

total number of outcomes

Ready to Go On? Problem Solving InterventionPermutations 11-7

LESSON



11-5 Probability of Independent and Dependent EventsDecide whether each set of events is independent ordependent. Explain your answer.

1. Alicia flips a dime and a penny. The dime comes up heads andthe penny comes up tails.

independent; Neither event has any connection with the other one.2. A box has ten names placed in it. First, Pedro draws a name randomly from

the box. Then, Kazuko randomly draws one of the remaining names.

dependent; The names Kazuko has to choose from will depend on the

result of Pedro’s selection.3. Graciela selects a card randomly from a full deck of playing

cards and looks at it before replacing it. Then, Richard carefullyshuffles the deck and selects a card.

independent; The first event does not influence the second event because

the selected card is replaced and the deck is shuffled.4. Abdul picks one of the ten digits randomly using his calculator.

Then, again using his calculator, he picks one of the nineremaining digits.

dependent; The nine digits available for the second selection will depend

on the result of the first selection because the selected digit will no

longer be available.

A hat has the names of 5 boys and 7 girls.

5. What is the probability of randomly selecting a girl from the names in the hat, replacing her name, and then randomly selecting a boy?

6. What is the probability of randomly selecting a boy from the names in the hat and then randomly selecting a girl without replacing the first name?

All but the 13 clubs have been removed from a deck of shuffledplaying cards.

7. What is the probability of drawing a face card and then another face card if there is no replacement?

8. What is the probability of drawing a face card and then another face card if there is replacement?

�1969�

�216�

�13352

�

�13454

�

Ready to Go On? Quiz11BSECTION


11-6 Combinations

9. An ice-cream shop has 10 different flavors. Frieda wants to buy an ice-cream cone with 2 scoops and different flavors. How many combinations of two different flavors are available?

10. Bill and 7 of his friends want to divide up into two teams to play some touch football. There will be 4 players on each team. In how many different ways can the members of the two teams be picked?

11. Mimi is buying board games to give her friends as gifts. There are 7 different games at the store. How many combinations of 3 different games are available?

12. Charles finds 9 different kinds of soup in the grocery store. He wants to buy 5 of these. How many combinations of 5 different kinds of soup are available?

13. Louisa and 5 of her friends want to divide up into two groups for a school project. There will be exactly 3 people in each group. In how many different ways can the members of the two groups be chosen?

11-7 Permutations

14. Bill’s touch football team has 4 players on it. In how many different ways can these 4 players be arranged in 4 different playing positions?

15. In how many different orders can Luann arrange 6 books on one of her bookshelves?

16. The school chess club has 9 members. Only 5 members at a time can occupy positions on the chess club team, which has a top board, second board, third board, and so on. In how many different ways can these 5 positions be filled?

17. Another school’s chess club has exactly 5 members. In how many different ways can the same 5 positions on the club chess team be filled?

18. The city is considering 8 students for 6 summer jobs. The 6 jobs are all different. How many ways are there of filling the 6 jobs? 20,160

120

15,120

720

24

20

126

35

70

45


11BSECTION


1. All but the 13 diamonds have been removed from a deck of playing cards. What is the probability of randomly drawing a face card (king, queen, or jack) from the deck?

2. Represent this on the array by shading the three horizontal rows K, Q, and J yellow. What fraction of the rows is shaded?

3. If the drawn card is replaced, what is the probability of randomly drawing a face card on a second draw?

4. Represent this by shading the three vertical columns K, Q, and J red. What fraction of the columns is shaded?

5. What is the probability of drawing a face card, replacing it, and then drawing a face card again?

6. What fraction of the small squares in the array is shaded both yellow and red?

7. Now compare your answers to questions 5 and 6. What do you notice about them?

8. If two cards are drawn without replacement from our deck of 13 cards, they cannot both be the same card. Draw an Xthrough each small square that represents two aces, two 2’s, two 3’s, and so on. How many squares have you crossed out?

9. Two cards are drawn without replacement. What is the probability that both are face cards?

10. Excluding all the squares that are crossed out, what fraction of the small squares are shaded both yellow and red?

11. What do you notice about your last two answers? They are the same.

�216�

�216�

13

They are the same.

�1969�

�1969�

�133�

�133�

�133�

�133�

Ready to Go On? EnrichmentProbability and Arrays11B

SECTION

A 2 3 4 5 6 7 8 9 10 J Q K

A

2

3

4

5

6

7

8

9

10

J

Q

K

To solve equations with more than one operation, or a two-stepequation, follow the order of operations in reverse. First add orsubtract and then multiply or divide.

Solving Two-Step Equations Using DivisionSolve. Check your answer.

7x � 9 � 37

What number do you subtract from both sides?

7x �

�77x� � �

278� What number do you divide by to isolate the variable?

x � What does x equal?

Check:

7x � 9 � 37

7( ) � 9 �?

37 Substitute for x into the equation.

� 9 �?

37 Multiply.

� 37 Does the answer check?

Solving Two-Step Equations Using MultiplicationSolve. Check your answer.

9 � �p8

� � 16

What number do you subtract from both sides?

�p8

� �

( ) �p8

� � (7) What number do you multiply by to isolate the variable?

p �

Check:

9 � �p8

� � 16

9 � �586� �

?16 Substitute for p into the equation.

9 � �?

16 Divide.

� 16 Does the answer check? yes167

56

56

88

7�9�9

yes3728

44

4

28�9�9


Ready to Go On? Skills InterventionSolving Two-Step Equations12-1

SECTION

Documents

LESSON Ready to Go On? Skills Intervention 11-1 Probability · she will not make tuna today? How many choices of sandwich are there? ... Skills Intervention 11-1 Probability LESSON