Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences

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372 7 PROBABILITY

APPLIED EXAMPLE 1

SAT Verbal Scores The superintendent of a

metropolitan school district has estimated the probabilities associated with

the SAT verbal scores of students from that district. The results are shown in

Table 7. If a student is selected at random, what is the probability that his or her

SAT verbal score will be

a. More than 400?

b. Less than or equal to 500?

c. Greater than 400 but less than or equal to 600?

Solution

Let A, B, C, D, E, and F denote, respectively, the event that the score

is greater than 700, greater than 600 but less than or equal to 700, greater than

500 but less than or equal to 600, and so forth. Then these events are mutually

exclusive. Therefore,

a. The probability that the student’s score will be more than 400 is given by

b. The probability that the student’s score will be less than or equal to 500 is

given by

c. The probability that the student’s score will be greater than 400 but less than

or equal to 600 is given by

Property 3 holds if and only if E and F are mutually exclusive. In the general case, we

have the following rule.

To derive this property, refer to Figure 8. Observe that we can write

as a union of disjoint sets. Therefore,

and

Finally, since E 傼 F  (E 傽 F

) 傼 (E 傽 F) 傼 (E

傽 F) is a union of disjoint sets, we

have

P1F2 P1E

艚 F 2 P1E 艚 F 2

P1E

艚 F 2 P1F2 P1E 艚 F2

P1E2 P1E 艚 F

2 P1E 艚 F2

P1E 艚 F

2 P1E2 P1E 艚 F2

E  1E 艚 F

2傼 1E 艚 F 2

and

F  1E

艚 F 2傼 1E 艚 F 2

Property 4. Addition Rule

If E and F are any two events of an experiment, then

 .19  .23  .42

P1C 傼 D 2 P1C 2 P1D 2

 .23  .31  .19  .73

P1D 傼 E 傼 F 2 P1D 2 P1E 2 P1F2

 .5

 .23  .19  .07  .01

P1D 傼 C 傼 B 傼 A 2 P1D 2 P1C 2 P1B2 P1A2

FIGURE 8

E 傼 F  (E 傽 F

) 傼 (E 傽 F) 傼

傽 F )

TABLE 7

Probability Distribution

Score, x Probability

x  700 .01

600  x  700 .07

500  x  600 .19

400  x  500 .23

300  x  400 .31

x  300 .19

P1E 傼 F2 P1E2 P1F2 P1E 艚 F 2

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 372

Note

Observe that if E and F are mutually exclusive—that is, if E 傽 F —

then the equation of Property 4 reduces to that of Property 3. In other words, if E

and F are mutually exclusive events, then P(E 傼 F )  P(E)  P(F ). If E and F are

not mutually exclusive events, then P(E 傼 F)  P(E)  P(F )  P(E 傽 F).

EXAMPLE 2

A card is drawn from a well-shuffled deck of 52 playing cards. What

is the probability that it is an ace or a spade?

Solution

Let E denote the event that the card drawn is an ace and let F denote the

event that the card drawn is a spade. Then,

Furthermore, E and F are not mutually exclusive events. In fact, E 傽 F is the event

that the card drawn is the ace of spades. Consequently,

The event that a card drawn is an ace or a spade is E 傼 F, with probability given by

(Figure 9). This result, of course, can be obtained by arguing that 16 of the 52 cards

are either spades or aces of other suits.

APPLIED EXAMPLE 3

Quality Control The quality-control depart-

ment of Vista Vision, manufacturer of the Pulsar 42-inch plasma TV, has

determined from records obtained from the company’s service centers that 3% of

the sets sold experience video problems, 1% experience audio problems, and 0.1%

experience both video and audio problems before the expiration of the 90-day

warranty. Find the probability that a plasma TV purchased by a consumer will

experience video or audio problems before the warranty expires.

Solution

Let E denote the event that a plasma TV purchased will experience

video problems within 90 days, and let F denote the event that a plasma TV pur-

chased will experience audio problems within 90 days. Then,

P(E)  .03 P(F )  .01 P(E 傽 F )  .001









P1E 傼 F 2 P1E 2 P1F 2 P1E 艚 F 2

P 1E 艚 F 2

P1E2

and

P1F2

 P1E2 P1F 2 P1E 艚 F 2

 P1E2 P1E 艚 F2 P1E 艚 F 2 P1F 2 P1E 艚 F 2

P1E 傼 F 2 P1E 艚 F

2 P1E 艚 F2 P1E

艚 F 2

7.3 RULES OF PROBABILITY 373

Aces

n(E F) = 1

Spades

Ace of spades

FIGURE 9

P(E 傼 F )  P(E)  P(F )  P(E 傽 F)

Use the earlier

results.

Explore & Discuss

Let E, F, and G be any three events of an experiment. Use Formula (5) of Section 6.2 to

show that

If E, F, and G are pairwise mutually exclusive, what is P(E 傼 F 傼 G)?

 P 1F 艚 G 2 P 1E 艚 F 艚 G 2

P1E 傼 F 傼 G2 P1E 2 P1F2 P1G2 P1E 艚 F2 P1E 艚 G 2

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 373

The event that a plasma TV purchased will experience video problems or audio

problems before the warranty expires is E 傼 F, and the probability of this event

is given by

(Figure 10).

Here is another property of a probability function that is of considerable aid in

computing the probability of an event.

Property 5 is an immediate consequence of Properties 2 and 3. Indeed, we have

E 傼 E

 S and E 傽 E

, so

and, therefore,

P1E

2 1  P1E2

1  P1S2 P 1E 傼 E

2 P1E2 P1E

Property 5. Rule of Complements

If E is an event of an experiment and E

denotes the complement of E, then

P(E

)  1  P(E)

 .039

 .03  .01  .001

P1E 傼 F2 P1E2 P1F2 P1E 艚 F 2

374 7 PROBABILITY

P(E) = .03

P(F) = .01

F) = .001

P(E

FIGURE 10

P(E 傼 F )  P(E)  P(F )  P(E 傽 F )

PORTFOLIO

The insurance business is all

about probabilities. Maybe your car

will be stolen, maybe it won’t;

maybe you’ll be in an accident,

maybe you won’t; maybe someone will slip on your steps

and sue you; maybe they won’t. Naturally, we hope that

nothing bad happens to you or your family, but common

sense and historical statistics tell us that it might.

All these have probabilities associated with them. For

example, if you own a Toyota Camry, there is a much

higher probability your car will be stolen than if you drive

a Dodge Caravan. As a matter of fact, in recent statistics

listing the top ten stolen cars in America, the Toyota

Camry was not only in first place; various model years

were also in second, third, and eighth place! And the

Honda Accord was also pretty popular with thieves—

during the same year, different model years of the Accord

were in fifth, seventh, ninth, and tenth place on the top-

ten stolen list.

In terms of safety statistics, we’ve found that accidents

and injuries tend to vary quite a bit from one make and

model to another. With other factors being equal, a person

who drives a Pontiac Firebird is much more likely to be

involved in an accident than someone who drives a Buick

Park Avenue. For example, a 23-year-old, unmarried male

with a good driving record who uses his Firebird for com-

muting and pleasure driving in Philadelphia, Pennsylvania,

would be charged an annual insurance premium over 30%

higher than the same person driving a Buick Park Avenue

would pay for the same personal injury and liability coverage.

Naturally, we at Good and Associates wish you nothing

but the best in terms of avoiding

crime, staying healthy, and being acci-

dent free. However, we are glad to be

here to help you prepare for negative

possibilities that might come your

way and to help you deal with them

if they do.

Todd Good

TITLE Owner and Broker

INSTITUTION Good and Associates

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 374

APPLIED EXAMPLE 4

Warranties Refer to Example 3. What is the

probability that a Pulsar 42-inch plasma TV bought by a consumer will not

experience video or audio difficulties before the warranty expires?

Solution

Let E denote the event that a plasma TV bought by a consumer will

experience video or audio difficulties before the warranty expires. Then, the event

that the plasma TV will not experience either problem before the warranty

expires is given by E

, with probability

Computations Involving the Rules of Probability

We close this section by looking at two additional examples that illustrate the rules of

probability.

EXAMPLE 5

Let E and F be two mutually exclusive events and suppose that

P(E)  .1 and P(F)  .6. Compute:

a. P(E 傽 F) b. P(E 傼 F ) c. P(E

)

d. P(E

傽 F

) e. P(E

傼 F

)

Solution

a. Since the events E and F are mutually exclusive—that is, E 傽 F —we have

P(E 傽 F)  0.

b. P(E 傼 F )  P(E )  P(F )

Since

and

are mutually exclusive

 .1  .6

 .7

c. P(E

)  1  P(E) Property 5

 1  .1

 .9

d. Observe that, by De Morgan’s law, E

傽 F

 (E 傼 F)

. Hence,

P(E

傽 F

)  P[(E 傼 F)

] See Figure 11.

 1  P(E 傼 F) Property 5

 1  .7 Use the result of part (b).

 .3

e. Again using De Morgan’s law, we find

P(E

傼 F

)  P[(E 傽 F)

]

 1  P(E 傽 F)

 1  0 Use the result of part (a).

 1

EXAMPLE 6

Let E and F be two events of an experiment with sample space S.

Suppose P(E)  .2, P(F)  .1, and P(E 傽 F )  .05. Compute:

a. P(E 傼 F )

b. P(E

傽 F

)

c. P(E

傽 F )

Hint:

Draw a Venn diagram.

 .961

 1  .039

P1E

2 1  P1E2

7.3 RULES OF PROBABILITY 375

E F

P(E) = .1 P(F) = .6

FIGURE 11

P(E

傽 F

)  P[(E 傼 F )

]

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 375

Solution

a. P(E 傼 F )  P(E )  P(F )  P(E 傽 F ) Property 4

 .2  .1  .05

 .25

b. Using De Morgan’s law, we have

P(E

傽 F

)  P[(E 傼 F)

]

 1  P(E 傼 F) Property 5

 1  .25 Use the result of part (a).

 .75

c. From the Venn diagram describing the relationship among E, F, and S (Figure 12),

we have

P(E

傽 F )  .05 The shaded subset is the event

傽

This result may also be obtained by using the relationship

P(E

傽 F )  P(F )  P(E 傽 F )

 .1  .05

 .05

376 7 PROBABILITY

P(F) = .1

.05.15

P(E) = .2

.05

FIGURE 12

P(E

傽 F ): the probability that the event

F, but not the event E, will occur

7.3 Exercises

7.3 Self-Check Exercises

7.3 Concept Questions

1. Let E and F be events of an experiment with sample space

S. Suppose P(E)  .4, P(F)  .5, and P(E 傽 F)  .1.

Compute:

a. P(E 傼 F) b. P(E 傽 F

)

2. Susan Garcia wishes to sell or lease a condominium

through a realty company. The realtor estimates that the

probability of finding a buyer within a month of the date

the property is listed for sale or lease is .3, the probability

of finding a lessee is .8, and the probability of finding both

a buyer and a lessee is .1. Determine the probability that

the property will be sold or leased within 1 month from the

date the property is listed for sale or lease.

Solutions to Self-Check Exercises 7.3 can be found on

page 381.

1. Suppose S is a sample space of an experiment, E and F are

events of the experiment, and P is a probability function.

Give the meaning of each of the following statements:

a. P(E)  0 b. P(F)  0.5 c. P(S)  1

d. P(E 傼 F)  P(E)  P(F)  P(E 傽 F)

2. Give an example, based on a real-life situation, illustrating

the property P(E

)  1  P(E), where E is an event and E

is the complement of E.

A pair of dice is rolled, and the number that appears

uppermost on each die is observed. In Exercises 1–6,

refer to this experiment and find the probability of the

given event.

1. The sum of the numbers is an even number.

2. The sum of the numbers is either 7 or 11.

3. A pair of 1s is thrown.

4. A double is thrown.

5. One die shows a 6, and the other is a number less than 3.

6. The sum of the numbers is at least 4.

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 376

7.3 RULES OF PROBABILITY 377

An experiment consists of selecting a card at random

from a 52-card deck. In Exercises 7–12, refer to this

experiment and find the probability of the event.

7. A king of diamonds is drawn.

8. A diamond or a king is drawn.

9. A face card (i.e., a jack, queen, or king) is drawn.

10. A red face card is drawn.

11. An ace is not drawn.

12. A black face card is not drawn.

13. Five hundred raffle tickets were sold. What is the proba-

bility that a person holding one ticket will win the first

prize? What is the probability that he or she will not win

the first prize?

14. TV H

OUSEHOLDS

The results of a recent television survey

of American TV households revealed that 87 out of every

100 TV households have at least one remote control. What

is the probability that a randomly selected TV household

does not have at least one remote control?

In Exercises 15–24, explain why the statement is incorrect.

15. The sample space associated with an experiment is given by

S  {a, b, c}, where P(a)  .3, P(b)  .4, and P(c)  .4.

16. The probability that a bus will arrive late at the Civic Cen-

ter is .35, and the probability that it will be on time or early

is .60.

17. A person participates in a weekly office pool in which he

has one chance in ten of winning the purse. If he partici-

pates for 5 weeks in succession, the probability of winning

at least one purse is .

18. The probability that a certain stock will increase in value

over a period of 1 week is .6. Therefore, the probability

that the stock will decrease in value is .4.

19. A red die and a green die are tossed. The probability that a

6 will appear uppermost on the red die is , and the proba-

bility that a 1 will appear uppermost on the green die is

. Hence, the probability that the red die will show a 6 or

the green die will show a 1 is .

20. Joanne, a high school senior, has applied for admission to

four colleges, A, B, C, and D. She has estimated that the

probability that she will be accepted for admission by A, B,

C, and D is .5, .3, .1, and .08, respectively. Thus, the prob-

ability that she will be accepted for admission by at least

one college is P(A)  P(B)  P(C )  P(D)  .5  .3 

.1  .08  .98.

21. The sample space associated with an experiment is given

by S  {a

, b, c, d, e}. The events E  {a, b} and F  {c, d}

are mutually exclusive. Hence, the events E

and F

are

mutually exclusive.



22. A 5-card poker hand is dealt from a 52-card deck. Let A

denote the event that a flush is dealt, and let B be the event

that a straight is dealt. Then the events A and B are mutu-

ally exclusive.

23. R

ETAIL

ALES

Mark Owens, an optician, estimates that the

probability that a customer coming into his store will pur-

chase one or more pairs of glasses but not contact lenses is

.40, and the probability that he will purchase one or more

pairs of contact lenses but not glasses is .25. Hence, Owens

concludes that the probability that a customer coming into

his store will purchase neither a pair of glasses nor a pair

of contact lenses is .35.

24. There are eight grades in Garfield Elementary School. If a

student is selected at random from the school, then the

probability that the student is in the first grade is .

25. Let E and F be two events that are mutually exclusive, and

suppose P(E )  .2 and P(F )  .5. Compute:

a. P(E 傽 F) b. P(E 傼 F )

c. P(E

) d. P(E

傽 F

)

26. Let E and F be two events of an experiment with sample

space S. Suppose P(E)  .6, P(F)  .4, and P(E 傽 F) 

.2. Compute:

a. P(E 傼 F) b. P(E

)

c. P(F

) d. P(E

傽 F)

27. Let S  {s

, s

} be the sample space associated with

an experiment having the probability distribution shown in

the accompanying table. If A  {s

, s

} and B  {s

, s

find

a. P(A), P(B) b. P(A

), P(B

)

c. P(A 傽 B) d. P(A 傼 B)

Outcome Probability

28. Let S  {s

, s

} be the sample space associ-

ated with an experiment having the probability distribu-

tion shown in the accompanying table. If A  {s

, s

} and

B  {s

, s

}, find

a. P(A), P(B) b. P(A

), P(B

)

c. P(A 傽 B) d. P(A 傼 B)

e. P(A

傽 B

) f. P(A

傼 B

)

Outcome Probability

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 377

29. T

EACHER

TTITUDES

A nonprofit organization conducted a

survey of 2140 metropolitan-area teachers regarding their

beliefs about educational problems. The following data

were obtained:

900 said that lack of parental support is a problem.

890 said that abused or neglected children are problems.

680 said that malnutrition or students in poor health is a

problem.

120 said that lack of parental support and abused or ne-

glected children are problems.

110 said that lack of parental support and malnutrition or

poor health are problems.

140 said that abused or neglected children and malnutrition

or poor health are problems.

40 said that lack of parental support, abuse or neglect, and

malnutrition or poor health are problems.

What is the probability that a teacher selected at random

from this group said that lack of parental support is the

only problem hampering a student’s schooling?

Hint: Draw a Venn diagram.

30. I

NVESTMENTS

In a survey of 200 employees of a company

regarding their 401(k) investments, the following data

were obtained:

141 had investments in stock funds.

91 had investments in bond funds.

60 had investments in money market funds.

47 had investments in stock funds and bond funds.

36 had investments in stock funds and money market

funds.

36 had investments in bond funds and money market

funds.

22 had investments in stock funds, bond funds, and

money market funds.

What is the probability that an employee of the company

chosen at random

a. Had investments in exactly two kinds of investment

funds?

b. Had investments in exactly one kind of investment

fund?

c. Had no investment in any of the three types of funds?

31. R

ETAIL

ALES

The probability that a shopper in a certain

boutique will buy a blouse is .35, that she will buy a pair of

pants is .30 and that she will buy a skirt is .27. The proba-

bility that she will buy both a blouse and a skirt is .15, that

she will buy both a skirt and a pair of pants is .19, and that

she will buy both a blouse and a pair of pants is .12.

Finally, the probability that she will buy all three items is

.08. What is the probability that a customer will buy

a. Exactly one of these items?

b. None of these items?

32. C

OURSE

NROLLMENTS

Among 500 freshmen pursuing a

business degree at a university, 320 are enrolled in an eco-

nomics course, 225 are enrolled in a mathematics course,

and 140 are enrolled in both an economics and a mathe-

matics course. What is the probability that a freshman

selected at random from this group is enrolled in

a. An economics and/or a mathematics course?

b. Exactly one of these two courses?

c. Neither an economics course nor a mathematics course?

33. C

ONSUMER

URVEYS

A leading manufacturer of kitchen

appliances advertised its products in two magazines: Good

Housekeeping and the Ladies Home Journal. A survey of

500 customers revealed that 140 learned of its products

from Good Housekeeping, 130 learned of its products from

the Ladies Home Journal, and 80 learned of its products

from both magazines. What is the probability that a person

selected at random from this group saw the manufacturer’s

advertisement in

a. Both magazines?

b. At least one of the two magazines?

c. Exactly one magazine?

34. R

OLLOVER

EATHS

The following table gives the number of

people killed in rollover crashes in various types of vehi-

cles in 2002:

Types of Vehicles Cars Pickups SUVs Vans

Deaths 4768 2742 2448 698

Find the empirical probability distribution associated with

these data. If a fatality due to a rollover crash in 2002 is

picked at random, what is the probability that the victim

was in

a. A car? b. An SUV? c. A pickup or an SUV?

Source: National Highway Traffic Safety Administration

35. T

REPARATION

A survey in which people were asked

how they were planning to prepare their taxes in 2007

revealed the following:

Method of

Preparation Percent

Computer software 33.9

Accountant 23.6

Tax preparation service 17.4

Spouse, friend, or other

relative will prepare 10.8

By hand 14.3

What is the probability that a randomly chosen participant

in the survey

a. Was planning to use an accountant or a tax preparation

service to prepare his taxes?

b. Was not planning to use computer software to prepare

his taxes and was not planning to do his taxes by hand?

Source: National Retail Federation

378 7 PROBABILITY

87533_07_ch7_p353-416 1/30/08 10:05 AM Page 378

36. W

OMEN

’

PPAREL

In an online survey for Talbots of 1095

women ages 35 yr and older, the participants were asked

what article of clothing women most want to fit perfectly.

A summary of the results of the survey follows:

Article of Clothing Respondents

Jeans 470

Black Pantsuit 307

Cocktail Dress 230

White Shirt 22

Gown 11

Other 55

If a woman who participated in the survey is chosen at ran-

dom, what is the probability that she most wants

a. Jeans to fit perfectly?

b. A black pantsuit or a cocktail dress to fit perfectly?

Source: Market Tool’s Zoom Panel

37. S

WITCHING

OBS

Two hundred workers were asked: Would

a better economy lead you to switch jobs? The results of

the survey follow:

Very Somewhat Somewhat Very Don’t

Answer likely likely unlikely unlikely know

Respondents 40 28 26 104 2

If a worker is chosen at random, what is the probability that

he or she

a. Is very unlikely to switch jobs?

b. Is somewhat likely or very likely to switch jobs?

Source: Accountemps

38. 401(

) I

NVESTORS

According to a study conducted in 2003

concerning the participation, by age, of 401(k) investors,

the following data were obtained:

Age 20s 30s 40s 50s 60s

Percent 11 28 32 22 7

a. What is the probability that a 401(k) investor selected at

random is in his or her 20s or 60s?

b. What is the probability that a 401(k) investor selected at

random is under the age of 50?

Source: Investment Company Institute

39. A

LTERNATIVE

NERGY

OURCES

In a poll conducted among

likely voters by Zogby International, voters were asked

their opinion on the best alternative to oil and coal. The

results are as follows:

Fuel Other/

Source Nuclear Wind cells Biofuels Solar no answer

Respondents, % 14.2 16.0 3.8 24.3 27.9 13.8

What is the probability that a randomly selected participant

in the poll mentioned

a. Wind or solar energy sources as the best alternative to

oil and coal?

b. Nuclear or biofuels as the best alternative to oil and

coal?

Source: Zogby International

40. E

LECTRICITY

ENERATION

Electricity in the United States is

generated from many sources. The following table gives

the sources as well as their share in the production of elec-

tricity:

Source Coal Nuclear Natural gas Hydropower Oil Other

Share, % 50.0 19.3 18.7 6.7 3.0 2.3

If a source for generating electricity is picked at random,

what is the probability that it comes from

a. Coal or natural gas?

b. Nonnuclear sources?

Source: Energy Information Administration

41. D

OWNLOADING

USIC

The following table, compiled in

2004, gives the percentage of music downloaded from the

United States and other countries by U.S. users:

Country U.S. Germany Canada Italy U.K. France Japan Other

Percent 45.1 16.5 6.9 6.1 4.2 3.8 2.5 14.9

a. Verify that the table does give a probability distribution

for the experiment.

b. What is the probability that a user who downloads

music, selected at random, obtained it from either the

United States or Canada?

c. What is the probability that a U.S. user who downloads

music, selected at random, does not obtain it from Italy,

the United Kingdom (U.K.), or France?

Source: Felix Oberholtzer-Gee and Koleman Strumpf

42. P

LANS TO

EEP

ARS

In a survey conducted to see how

long Americans keep their cars, 2000 automobile owners

were asked how long they plan to keep their present cars.

The results of the survey follow:

Years Car Is Kept, x Respondents

0  x  160

1  x  3 440

3  x  5 360

5  x  7 340

7  x  10 240

10  x 560

Find the probability distribution associated with these data.

What is the probability that an automobile owner selected

at random from those surveyed plans to keep his or her

present car

a. Less than 5 yr?

b. 3 yr or more?

7.3 RULES OF PROBABILITY 379

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43. A

SSEMBLY

-T

IME

TUDIES

A time study was conducted by

the production manager of Universal Instruments to deter-

mine how much time it took an assembly worker to com-

plete a certain task during the assembly of its Galaxy home

computers. Results of the study indicated that 20% of the

workers were able to complete the task in less than 3 min,

60% of the workers were able to complete the task in 4 min

or less, and 10% of the workers required more than 5 min

to complete the task. If an assembly-line worker is selected

at random from this group, what is the probability that

a. He or she will be able to complete the task in 5 min or

less?

b. He or she will not be able to complete the task within

4 min?

c. The time taken for the worker to complete the task will

be between 3 and 4 min (inclusive)?

44. D

ISTRACTED

RIVING

According to a study of 100 drivers in

metropolitan Washington D.C. whose cars were equipped

with cameras with sensors, the distractions and the number

of incidents (crashes, near crashes, and situations that

require an evasive maneuver after the driver was dis-

tracted) caused by these distractions are as follows:

Distraction ABCDEFGHI

Driving Incidents 668 378 194 163 133 134 111 111 89

where A  Wireless device (cell phone, PDA)

B  Passenger

C  Something inside car

D  Vehicle

E  Personal hygiene

F  Eating

G  Something outside car

H  Talking/singing

I  Other

If an incident caused by a distraction is picked at random,

what is the probability that it was caused by

a. The use of a wireless device?

b. Something other than personal hygiene or eating?

Source: Virginia Tech Transportation Institute and NHTSA

45. G

-C

ONTROL

AWS

A poll was conducted among 250 res-

idents of a certain city regarding tougher gun-control laws.

The results of the poll are shown in the table:

Own Own Own a

Only a Only a Handgun Own

Handgun Rifle and a Rifle Neither Total

Favor

Tougher Laws 0 12 0 138 150

Oppose

Tougher Laws 58 5 25 0 88

Opinion 00 0 1212

Total 58 17 25 150 250

If one of the participants in this poll is selected at random,

what is the probability that he or she

a. Favors tougher gun-control laws?

b. Owns a handgun?

c. Owns a handgun but not a rifle?

d. Favors tougher gun-control laws and does not own a

handgun?

46. R

ISK OF AN

IRPLANE

RASH

According to a study of

Western-built commercial jets involved in crashes from

1988 to 1998, the percentage of airplane crashes that occur

at each stage of flight are as follows:

Phase Percent

On ground, taxiing 4

During takeoff 10

Climbing to cruise altitude 19

En route 5

Descent and approach 31

Landing 31

If one of the doomed flights in the period 1988–1998 is

picked at random, what is the probability that it crashed

a. While taxiing on the ground or while en route?

b. During takeoff or landing?

If the study is indicative of airplane crashes in general,

when is the risk of a plane crash the highest?

Source: National Transportation Safety Board

47. Suppose the probability that Bill can solve a problem is p

and the probability that Mike can solve it is p

. Show that

the probability that Bill and Mike working independently

can solve the problem is p

 p

 p

48. Fifty raffle tickets are numbered 1 through 50, and one of

them is drawn at random. What is the probability that the

number is a multiple of 5 or 7? Consider the following

“solution”: Since 10 tickets bear numbers that are multi-

ples of 5 and since 7 tickets bear numbers that are multi-

ples of 7, we conclude that the required probability is

What is wrong with this argument? What is the correct

answer?

In Exercises 49–52, determine whether the statement is

true or false. If it is true, explain why it is true. If it is

false, give an example to show why it is false.

49. If A is a subset of B and P(B)  0, then P(A)  0.

50. If A is a subset of B, then P(A)  P(B).

51. If E

, E

,..., E

are events of an experiment, then

P(E

傼 E

傼

. . .

傼 E

)  P(E

)  P(E

) 

. . .

 P(E

52. If E is an event of an experiment, then P(E )  P(E

)  1.





380 7 PROBABILITY

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Further Applications of Counting Techniques

As we have seen many times before, a problem in which the underlying sample space

has a small number of elements may be solved by first determining all such sample

points. However, for problems involving sample spaces with a large number of sam-

ple points, this approach is neither practical nor desirable.

In this section, we see how the counting techniques studied in Chapter 6 may be

employed to help us solve problems in which the associated sample spaces contain

large numbers of sample points. In particular, we restrict our attention to the study of

uniform sample spaces—that is, sample spaces in which the outcomes are equally

likely. For such spaces we have the following result.

EXAMPLE 1

An unbiased coin is tossed six times. What is the probability that the

coin will land heads

a. Exactly three times?

b. At most three times?

c. On the first and the last toss?

Computing the Probability of an Event in a Uniform Sample Space

Let S be a uniform sample space and let E be any event. Then

(1)

7.4 USE OF COUNTING TECHNIQUES IN PROBABILITY 381

1. a. Using Property 4, we find

b. From the accompanying Venn diagram, in which the

subset E 傽 F

is shaded, we see that

P(E 傽 F

)  .3

.3 .1 .4

 .8

 .4  .5  .1

P1E 傼 F 2 P1E2 P1F 2 P1E 艚 F 2

The result may also be obtained by using the relationship

2. Let E denote the event that the realtor will find a buyer

within 1 month of the date it is listed for sale or lease and

let F denote the event that the realtor will find a lessee

within the same time period. Then,

P(E)  .3 P(F )  .8 P(E 傽 F )  .1

The probability of the event that the realtor will find a

buyer or a lessee within 1 month of the date it is listed for

sale or lease is given by

—that is, a certainty.

 .3  .8  .1  1

P1E 傼 F 2 P1E2 P1F 2 P1E 艚 F 2

 .4  .1  .3

P1E 艚 F

2 P 1E2 P1E 艚 F2

P1E2

Number of outcomes in E

Number of outcomes in S



n1E2

n1S2

7.3 Solutions to Self-Check Exercises

7.4 Use of Counting Techniques in Probability

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