8.8 Computation of Compound Probabilities - Cengage · 8.8 Computation of Compound Probabilities...

8.8 Computation of Compound Probabilities 8-1

8.8 Computation of Compound Probabilities

Objectives

1. Compute Probabilities of Compound Events2. Compute Probabilities of Independent Events

In the previous section, we discussed how to compute the probability of a singleevent. For example, the probability of being dealt the very rare poker hand, called a royal flush, is , about one in two and one-halfmillion.

In this section, we will discuss how to find the probability of one event oranother, or the probability of one event and another. Such events are calledcompound events.

1. Compute Probabilities of Compound EventsWe have seen that if and are two events, the probability that and will bothoccur is . In this section, we also discuss , the probability thateither or will occur.

Suppose we want to find the probability of drawing a king or a heart from astandard card deck. If is the event “drawing a king” and is the event “drawing a heart,” then , and . However, the probability of drawing a king or a heart is not the sum of these two probabilities. Because the king ofhearts was counted twice, once as a king and once as a heart, and because the probability of drawing the king of hearts is , we must subtract from the sumof and to get the correct probability.

In general, we have the following rule.

If and are two events, then

If events and have no outcomes in common, then and. Such events are called mutually exclusive (if one event

occurs, the other cannot).Here some examples of events that are mutually exclusive and some that are

not.

P(A � B) � P(�) � 0A � B � �BA

P(A � B) � P(A) � P(B) � P(A � B)

BAP(A � B)

�4

13

�1652

�452

�1352

�1

52

P(K � H) � P(K) � P(H) � P(K � H) P(king or heart) � P(king) � P(heart) � P(king of hearts)

1352

452

152

152

P(H) �1352P(K) �

452

HK

BAP(A � B)P(A � B)

BABA

1C(52,5) �

12,598,960

C(5, 5)C(52, 5) �

8-2 Chapter 8 Natural-Number Functions and Probability

Mutually exclusive events Non-mutually exclusive events

Giving birth to a boy or a girl Giving birth to a boy or a baby weighing more than 6 pounds

Rolling a 2 or 3 on one roll of a die Rolling a 2 or an even number on one roll of a die

Drawing an ace or a king on one draw Drawing an ace or a red card from a from a standard card deck standard card deck

The following rule applies to mutually exclusive events.

If and cannot occur simultaneously, then

P(A � B) � P(A) � P(B)

BAP(A � B)

The event (read as “not ”) contains all outcomes of the sample space thatare not elements of event . Because the events and are mutually exclusive,

Because either event or event must happen, . Thus,

Add to both sides.

This result gives another property of compound probabilities.

If is any event, then

EXAMPLE 1 A counselor tells a student that his probability of earning a grade of in algebrais and his probability of earning an is . Find the probability that the studentearns a or better.

Solution Because “earning a ” and “earning an ” are mutually exclusive, the probabilityof earning a or is given by

Note that and .

The probability that the student will receive a or better is

�1925

� 1 �6

25

P(C or better) � 1 � P(D � F)

C

�6

25

�15

�1

25

F) � 0P(D P(D � F) � P(D) � P(F)

FDFD

C

125F1

5

D

P(A) � 1 � P(A)

AP(A)

�P(A) P(A) � 1 � P(A) P(A) � P(A) � 1

P(A � A) � 1

P(A � A) � 1AA

P(A � A) � P(A) � P(A)

AAAAA

Commentis also known as the complement

of event .AA

The probability of earning a or better is .

Self Check 1 Find the probability that the student passes (earns a or better).

2. Compute Probabilities of Independent EventsIf two events do not influence each other, they are called independent events.

The events and are said to be independent events if and only if.

Here are some examples of events that are independent and some that are not.

Independent events Non-independent events

Drawing an ace and a king on two Drawing an ace and a king on two draws from a standard card deck draws from a standard card deck with replacement without replacement

A basketball player making ten free A women catching a cold and throws in a row developing a cough

A student scoring two A’s on two tests The chance it will rain and the chance the sidewalks get wet

In the previous section, we discussed the multiplication property for probabilities:

Substituting for in this property gives a formula for computingprobabilities of compound independent events.

If and are independent events, then

The event of “drawing an ace from a standard deck of cards” and the eventof “tossing heads” on one toss of a coin are independent events, because neither

event influences the other. Consequently,

EXAMPLE 2 The probability that a baseball player can get a hit is . Find the probability thatshe will get three hits in a row.

Solution Assume that the three times at bat are independent events: One time at bat doesnot influence her chances of getting a hit on another turn at bat. Because

, , and ,P(E3) �13P(E2) �

13P(E1) �

13

13

�1

26

�452

�12

P(drawing an ace and tossing heads) � P(drawing an ace) � P(tossing heads) P(A � B) � P(A) � P(B)

BA

P(A � B) � P(A) � P(B)

BAFormula for P(A � B)

P(B ƒ A)P(B)

P(A � B) � P(A) � P(B ƒ A)

P(B) � P(B ƒ A)BAIndependent Events

D

1925C



The probability that she will get three hits in a row is .

Self Check 2 The probability that another player can get a hit is . Find the probability that shewill get four hits in a row.

EXAMPLE 3 A die is rolled three times. Find the probability that the outcome is 6 on the firstroll, an even number on the second roll, and an odd prime number on the third roll.

Solution The probability of a 6 on any roll is . Because there are three even inte-gers represented on a die, the probability of rolling an even number is

.Since 3 and 5 are the only odd prime numbers on a die, the probability of

rolling an odd prime is .Because these three events are independent, the probability of the events hap-

pening in succession is the product of the probabilities:

Self Check 3 Find the probability that the outcome is five on the first roll, an odd number onthe second roll, and two on the third roll.

EXAMPLE 4 The probability that a drug will cure dandruff is . However, if the drug is used,

the probability that it will cause side effects is . Find the probability that a patientwho uses the drug will be cured and will suffer no side effects.

Solution The probability that the drug will cure dandruff is . The probability of

having side effects is . The probability that the patient will have no sideeffects is

Since these events are independent,

�5

48

�18

�56

� P(C) � P(E) P(cure and no side effects) � P(C � E)

P(E) � 1 � P(E) � 1 �16

�56

P(E) �16

P(C) �18

16

18

�1

36

�16

�12

�13

� P(6) � P(E) � P(O) P(six and even number and odd prime) � P(6 � E � O)

P(odd prime) � P(O) �26 �

13

P(even number) � P(E) �36 �

12

P(6) �16

14

127

P(E1 � E2 � E3) �13

�13

�13

�1

27


Self Check 4 Find the probability that the patient will be cured and suffer side effects.

Self Check Answers 1. 2. 3. 4. 148

172

1256

2425

Vocabulary and Concepts Fill in the blanks.1. A event is one event or another or one

event followed by another.2.3. If and are , then

.4. The event is read as “ .”5.6. Two events, and , are called independent events

when .7. If and are independent events, then

.8. If two events do not influence each other, they are

called events.

Practice Assume that you draw one card from a stan-dard card deck. Find the probability of each event.

9. Drawing a black card10. Drawing a jack11. Drawing a black card or an ace12. Drawing a red card or a face card

Assume that you draw two cards from a standard carddeck, without replacement. Find the probability of eachevent.13. Drawing two aces14. Drawing three aces15. Drawing a club and then another black card16. Drawing a heart and then a spade

Assume that you roll two dice once. Find theprobability of each result.17. Rolling a sum of 7 or 618. Rolling a sum of 5 or an even sum19. Rolling a sum of 10 or an odd sum20. Rolling a sum of 12 or 1

P(A � B) �BA

BA

P(A) �

AP(A � B) � P(A) � P(B)

BA

P(A � B) �

Assume that you have a bucket containing 7 beigecapsules, 3 blue capsules, and 6 green capsules. Youmake a single draw from the bucket, taking onecapsule. Find the probability of each result.21. Drawing a beige or a blue capsule22. Drawing a green capsule23. Not drawing a blue capsule24. Not drawing either a beige or a blue capsule

Assume that you are using the same bucket of capsulesas in Exercises 21–24.25. On two draws from the bucket, find the probabil-

ity of drawing a beige capsule followed by a greencapsule. (Assume that the capsule is returned tothe bucket after the first draw.)

26. On two draws from the bucket, find the probabil-ity of drawing one blue capsule and one greencapsule. (Assume that the capsule is not returnedto the bucket after the first draw.)

27. On three successive draws from the bucket(without replacement), find the probability offailing to draw a beige capsule.

28. Jeff rolls a die and draws one card from a carddeck. Find the probability of his rolling a 4 anddrawing a four.

29. Birthday problem Three people are in an eleva-tor together. Find the probability that all threewere born on the same day of the week.

30. Birthday problem Three people are on a bustogether. Find the probability that at least onewas born on a different day of the week from theothers.

31. Birthday problem Five people are in a roomtogether. Find the probability that all five wereborn on a different day of the year.

32. Birthday problem Five people are on a bustogether. Find the probability that at least two ofthem were born on the same day of the year.

8.8 Exercises


33. Sharing homework If the probability that Rick will solve a problem is and the probability that

Dinah will solve it is , find the probability that at

least one of them will solve it.34. Signaling A bugle is used for communication at

camp. The call for dinner is based on four pitchesand is five notes long. If a child can play thesefour pitches on a bugle, find the probability thatthe first five notes that the child plays will call thecamp to dinner. (Assume that the child is equallylikely to play any of the four pitches each time anote is blown.)

A woman visits her cabin in Canada. The probability that her lawnmower will start is , the probability that

her gas-powered saw will start is , and the probability

that her outboard motor will start is . Find each probability.35. That all three will start36. That none will start37. That exactly one will start38. That exactly two will start

Applications39. Immigration Three children leave Thailand to

start a new life in either the United States orFrance. The probability that May Xao will go to France is , that Tou Lia will go to France is ,

and that May Moua will go to France is . Find the probability that exactly two of them will endup in the United States.

16

12

13

34

13

12

25

14

40. Preparing for the GED The administrators of aprogram to prepare people for the high schoolequivalency exam have found that 80% of the stu-dents require tutoring in math, 60% need help inEnglish, and 45% need work in both math andEnglish. Find the probability that a studentselected at random needs help with either math orEnglish.

41. Insurance losses The insurance underwritershave determined that in any one year, the proba-bility that George will have a car accident is 0.05,and that if he has an accident, the probabilitythat he will be hospitalized is 0.40. Find the prob-ability that George will have an accident but notbe hospitalized.

42. Grading homework One instructor grades home-work by randomly choosing 3 out of the 15 prob-lems assigned. Bill did only 8 problems. What isthe probability that he won’t get caught?

Discovery and Writing43. Explain the difference between dependent and

independent events, and give examples of each.44. Explain why a. and

b. .

Review Maximize subject to the given constraints.45. 46.

μx � 0y � 0y � 2x � 2x � y � 3

μx � 0y � 0x � 4x � 2y � 8

P � 3x � yP � 2x � yP

P(A ƒ A) � 0P(A ƒ A) � 1

8.9 Odds and Mathematical Expectation 8-7

8.9 Odds and Mathematical Expectation

Objectives

1. Define and Compute Odds2. Define and Compute Mathematical Expectation

At the horse races, you will often hear phrases such as

Sweet Pea is a long shot with odds of 30 to 1.

or

The odds on Barbaro are 8 to 5.

Since the concept of odds is closely related to the concept of probability, thepayoffs on bets at the track are based on the odds for each horse in a race.

When we say that the odds on Sweet Pea are 30 to 1, we generally mean thatthe odds against the horse are 30 to 1. As we will soon see, this is equivalent to saying that the probability of Sweet Pea winning is . When we say thatthe odds on Barbaro are 8 to 5, we mean that the probability of Barbaro winningis .

We will begin this section by discussing odds.

1. Define and Compute Odds

The odds for an event is the probability of a favorable outcome divided by theprobability of an unfavorable outcome.

The odds against an event is the probability of an unfavorable outcome dividedby the probability of a favorable outcome.

Odds

58 � 5 �

513

130 � 1 �

131

EXAMPLE 1 The probability that a horse will win a race is . Find the odds for and the oddsagainst the horse.

Solution Because the probability that the horse will win is , the probability that the horsewill not win is . The odds for the horse are

or 1 to 3. The odds against the horse are

or 3 to 1.

probability of a lossprobability of a win

�

3414

�31

� 3

probability of a winprobability of a loss

�

1434

�13

34

14

14

CommentNote that the odds for an event isthe reciprocal of the odds againstthe event.


Self Check 1 The probability that a horse will lose a race is . Find the odds for and the oddsagainst the horse.

Another way to find the odds in favor of an event is to divide the number offavorable outcomes by the number of unfavorable outcomes. In Example 1, this is .

To find the odds against an event, divide the number of unfavorable outcomesby the number of favorable outcomes. In Example 1, this is .

EXAMPLE 2 The odds against a horse are 30 to 1. Find: a. the probability that the horse willlose b. the probability that the horse will win

Solution If is the probability that the horse will lose, then is the the probability thatthe horse will win. By definition, we have

Multiply both sides by .

Add to both sides.

Divide both sides by 31.

The probability that the horse will lose is . The probability that the horse willwin is .

Self Check 2 The odds against a horse are 8 to 5. Find the probability that the horse will win.

2. Define and Compute Mathematical ExpectationSuppose we have a chance to play a simple game with the following rules:

1. Roll a single die once.2. If a 6 appears, win $3.3. If a 5 appears, win $1.4. If any other number appears, win .5. The cost to play (one roll of the die) is $1.

In this game, the probability of any one of the six outcomes—rolling a 6, 5 4,3, 2, or 1—is , and the winnings are $3, $1, and . The expected winnings canbe found by using the following equation and simplifying the right side:

� 1

�16

(6)

�16

(3 � 1 � 0.50 � 0.50 � 0.50 � 0.50)

E �16

(3) �16

(1) �16

(0.50) �16

(0.50) �16

(0.50) �16

(0.50)

E50¢16

50¢

1 �3031 �

131

3031

p �3031

30p 31p � 301 � p p � 30 � 30p

301

�p

1 � p

Odds against an event �probability of losing

probability of winning

1 � pp

31

13

15

Over the long run, we could expect to win $1 with every play of the game.Since it costs $1 to play the game, the expected gain or loss is 0. Because theexpected winnings are equal to the admission price, the game is fair.

If a certain event has different outcomes with probabilities , , , . . . ,and the winnings assigned to each outcome are , , , . . . , the

expected winnings, or mathematical expectation, is given by

EXAMPLE 3 It costs $1 to play the following game: Roll two dice; collect $5 if we roll a sum of7, and collect $2 if we roll a sum of 11. All other numbers pay nothing. Is it wiseto play the game?

Solution The probability of rolling a 7 on a single roll of two dice is , the probability of

rolling an 11 is , and the probability of rolling something else is . The mathe-

matical expectation is

By playing the game for a long period of time, we can expect to get backabout for every dollar spent. For the fun of playing, the cost is about agame. If the game is enjoyable, it might be worth the expected loss. However, thegame is slightly unfair.

Self Check 3 Is it wise to play the game if you win $4 when you roll a sum of 7 and $6 when youroll a sum of 11?

Self Check Answers 1. 4 to 1; 1 to 4 2. 3. Yes, the game is fair.513

5¢95¢

E �6

36(5) �

236

(2) �2836

(0) �1718

� $0.944

2836

236

636

E � P1x1 � P2x2 � P3x3 � p � Pnxn

Exnx3x2x1Pn

P3P2P1nMathematicalExpectation

8.9 Odds and Mathematical Expectation 8-9

Vocabulary and Concepts Fill in the blanks.1. The are the probability of a favor-

able outcome divided by the probability of anunfavorable outcome.

2. The are the probability of anunfavorable outcome divided by the probability ofa favorable outcome.

3. If the odds for an event are 1 to 4, the probabilityof winning is .

4. Mathematical expectation is given by .

Practice5. The probability that a horse will win a race is .

Find the odds for and against the horse.

15

E �E

6. The probability that a horse will win a race is .

Find the odds for and against the horse.

7. The odds against a horse are 50 to 1. Find theprobability that the horse will win.

8. The odds against a horse are 7 to 5. Find theprobability that the horse will lose.

Assume a single roll of a die.9. Find the probability of rolling a 6.

10. Find the odds in favor of rolling a 6.11. Find the odds against rolling a 6.12. Find the probability of rolling an even number.13. Find the odds in favor of rolling an even number.

23

8.9 Exercises

14. Find the odds against rolling an even number.

Assume a single roll of two dice.15. Find the probability of rolling a sum of 6.16. Find the odds in favor of rolling a sum of 6.

17. Find the odds against rolling a sum of 6.18. Find the probability of rolling an even sum.19. Find the odds in favor of rolling an even sum.

20. Find the odds against rolling an even sum.

Assume that you are drawing one card from a standardcard deck.21. Find the odds in favor of drawing a queen.

22. Find the odds against drawing a black card.

23. Find the odds in favor of drawing a face card.

24. Find the odds against drawing a diamond.25. If the odds in favor of victory are 5 to 2, find the

probability of victory.26. If the odds in favor of victory are 5 to 2, find the

odds against victory.27. If the odds against winning are 90 to 1, find the

odds in favor of winning.28. Find the odds in favor of rolling a 7 on a single

roll of two dice.29. Find the odds against tossing four heads in a row

with a fair nickel.30. Find the odds in favor of a couple having four

girl babies in succession.

31. The odds against a horse are 8 to 1. Find theprobability that the horse will win.

32. The odds against a horse are 1 to 1. Find theprobability that the horse will lose.

33. It costs $2 to play the following game:a. Draw one card from a card deck.b. Collect $5 if an ace is drawn.c. Collect $4 if a king is drawn.d. Collect nothing for all other cards drawn.Is it wise to play this game? Explain.

34. Lottery tickets One thousand tickets are soldfor a lottery with two grand prizes of $800. Finda fair price for the tickets.

1Assume P(girl) �12. 2

35. Find the odds against a couple having three baby boys in a row.

36. Find the odds in favor of tossing at least threeheads in five tosses of a fair coin.

37. Suppose you toss a coin five times and collect $5if you toss five heads, $4 if you toss four heads,$3 if you toss three heads, and no money for anyother combination. How much should you pay toplay the game if the game is to be fair?

38. If you roll two dice one time and collect $10 fordouble 6’s and $1 for double 1’s, what is a fairprice for playing the game?

39. Counting an ace as 1, a face card as 10, and allothers at their numerical values, find the expectedvalue if you draw one card from a card deck.

40. Find the expected sum of one roll of two dice.41. A multiple-choice test of eight questions gives

five possible answers for each question. Only oneof the answers for each question is right. Find theprobability of getting seven right answers bysimple guessing.

42. In the situation described in Exercise 41, find theodds in favor of getting seven answers right.

Discovery and Writing43. If “the odds are against an event,” what can be

said about its probability?44. To disguise the unlikely chance of winning, a

contest promoter publishes the odds in favor ofwinning as 0.0000000372 to 1. What are the oddsagainst winning?

Review45. Find the equation of the line perpendicular to

and passing through the point .

46. Find the equations of the parabolas with vertexat the point and passing through theorigin.

47. Find the equation of the circle with center atand tangent to the -axis.

48. Write the equation

in standard form, and identify the curve.

2x2 � 4y2 � 12x � 24y � 46 � 0

x(3, 5)

(3, �2)

(1, 1)3x � 2y � 9

1Assume P(boy) �12. 2


Chapter Review 8-11Chapter Review 8-11

CHAPTER REVIEW

8.8 Computation of Compound Probabilities

Definitions and Concepts Examples

If and are two events, then

P(A � B) � P(A) � P(B) � P(A � B)

BA Find the probability of drawing an ace or a heart onone draw from a standard card deck.

The probability of drawing an ace is .

The probability of drawing a heart is .

The ace of hearts has been counted both as an aceand as a heart. To get rid of the duplication, we mustsubtract the probability of getting the ace of heartsfrom the sum of the two probabilities listed above.

1352

�14

452

�1

13

�4

13

�1652

�113

�14

�1

52

P(ace or a heart) � P(ace) � P(heart) � P(ace of hearts)

If and cannot occur simultaneously, then

P(A � B) � P(A) � P(B)

BA Find the probability of drawing a 2 or a 3 from astandard card deck.

The probability of drawing a 2 is .

The probability of drawing a 3 is .

Since there is no duplication,

�2

13

�113

�1

13

P(2 or 3) � P(2) � P(3)

452

�1

13

452

�1

13

If is any event, then

P(A) � 1 � P(A)

A If the probability that a horse will win a race is , theprobability that the horse will lose the race is

.1 �7

11 �4

11

711

8-12 Chapter 8 Natural-Number Functions and Probability8-12 Chapter 8 Natural-Number Functions and Probability8-12 Chapter 8 Natural-Number Functions and Probability

If and are independent events, then

P(A � B) � P(A) � P(B)

BA A basketball player makes 80% of her free throws.Find the probability that she will make 2 free throwsin a row.

The probability that she makes 1 free throw is.

So the probability that she will make 2 in a row is

�45

�45

�1625

� P(1 free throw) � P(another free throw)P(1 free throw � another free throw)

80% �45

Exercises

83. If the probability that a drug cures a disease is0.83, and we give the drug to 800 people with thedisease, find the expected number of people whowill not be cured.

82. On one draw from a standard card deck, find theprobability of drawing a two or a spade.

8.9 Odds and Mathematical Expectation

Definitions and Concepts Examples

The odds for an event is the probability of a favorableoutcome divided by the probability of an unfavorableoutcome.

The odds against an event is the probability of anunfavorable outcome divided by the probability of afavorable outcome.

The probability that a politician will be elected is esti-mated to be 0.7. Find the odds for and against thepolitician.

Odds for the politician or 7 to 3.

Odds against the politician or 3 to 7.�0.30.7

�37

�0.70.3

�73

A game is fair if the cost to play equals the expectedwinnings .E

Suppose it costs $5 to play the following game:

Roll two dice: Collect $10 if you roll a 12 or a 2.Otherwise collect $1.

Is it wise to play the game?

Since the probability of rolling a 12 is , of rolling a

2 is , and of rolling something else is , the mathematical expectation is

Since it costs $5 to play and the expected earnings areonly $1.50, this is a foolish game to play.

E �1

36(10) �

136

(10) �3436

(1) � 1.5

3436

136

136

Chapter Review 8-13Chapter Review 8-13

Exercises

84. Find the odds against a horse if the probability that the horse will win is .

85. Find the odds in favor of a couple having 4 babygirls in a row.

86. If the probability that Joe will marry is and the

probability that John will marry is , find the odds against either one becoming a husband.

87. If the odds against Priscilla’s graduation from college are , find the probability that she will graduate.

1011

34

56

78

88. Find the expected earnings if you collect $1 forevery heads you get when you toss a fair coin4 times.

89. If the total number of subsets that a set with elements can have is , explain why

annb � 2nan

2b � p �an

1b �an

0b �

2nn

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