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

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46. S

EATING

RRANGEMENTS

In how many ways can seven stu-

dents be assigned seats in a row containing seven desks if

a. There are no restrictions?

b. Two of the students must not be seated next to each

other?

47. Q

UALITY

ONTROL

From a shipment of 60 CPUs, 5 of

which are defective, a sample of 4 CPUs is selected at

random.

a. In how many different ways can the sample be selected?

b. How many samples contain 3 defective CPUs?

c. How many samples do not contain any defective CPUs?

48. R

ANDOM

AMPLES

A sample of 4 balls is to be selected at

random from an urn containing 15 balls numbered 1 to 15.

If 6 balls are green, 5 are white, and 4 are black, then:

a. How many different samples can be selected?

b. How many samples can be selected that contain at least

1 white ball?

49. S

EATING

RRANGEMENTS

In how many ways can three mar-

ried couples be seated in a row of six seats

a. If there are no seating restrictions?

b. If men and women must alternate?

c. If each married couple must sit together?

50. T

EAM

ELECTION

There are seven boys and five girls in the

debate squad at Franklin High School. In how many ways

can a four-member debating team be selected

a. If there are no restrictions?

b. If two boys and two girls must be on the team?

c. If at least two boys must be on the team?

352

6 SETS AND COUNTING

Before Moving On . . .

CHAPTER 6

1. Let U  {a, b, c, d, e, f, t}, A  {a, d, f, t}, B  {d, f, t}, and

C  {b, c, e, f }. Find

a. A 傽 (B 傼 C)

b. (A 傽 C) 傼 (B 傼 C)

c. A

2. Let A, B, and C be subsets of a universal set U and sup-

pose that n(U)  120, n(A)  20, n(A 傽 B)  10,

n(A 傽 C)  11, n(B 傽 C)  9, and n(A 傽 B 傽 C)  4.

Find n[A 傽 (B 傼 C)

3. In how many ways can four compact discs be selected from

six different compact discs?

4. From a standard 52-card deck, how many 5-card poker

hands can be dealt consisting of 3 deuces and 2 face cards?

5. There are six seniors and five juniors in the Chess Club at

Madison High School. In how many ways can a team con-

sisting of three seniors and two juniors be selected from the

members of the Chess Club?

87533_06_ch6_p313-352 1/30/08 10:01 AM Page 352

What is the probability that a

randomly chosen couple from a

certain large metropolitan area is in

the upper-income bracket if both

spouses are working? In Example 2,

page 404, we show how the income

distribution for that metropolitan

area can be used to determine this

probability.

HE SYSTEMATIC STUDY of probability began in the 17th

century when certain aristocrats wanted to discover

superior strategies to use in the gaming rooms of Europe. Some

of the best mathematicians of the period were engaged in this

pursuit. Since then, probability has evolved in virtually every

sphere of human endeavor in which an element of uncertainty is

present.

We begin by introducing some of the basic terminology

used in the study of the subject. Then, in Section 7.2, we give

the technical meaning of the term probability. The rest of this

chapter is devoted to the development of techniques for

computing the probabilities of the occurrence of certain events.

PROBABILITY

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 353

354 7 PROBABILITY

Terminology

A number of specialized terms are used in the study of probability. We begin by defin-

ing the term experiment.

The results of the experiment are called the outcomes of the experiment. Three exam-

ples of experiments are the following:

■

Tossing a coin and observing whether it falls heads or tails

■

Rolling a die and observing which of the numbers 1, 2, 3, 4, 5, or 6 shows up

■

Testing a spark plug from a batch of 100 spark plugs and observing whether or not

it is defective

In our discussion of experiments, we use the following terms:

The sample space of an experiment is a universal set whose elements are precisely

the outcomes, or the sample points, of the experiment; the events of the experiment are

the subsets of the universal set. A sample space associated with an experiment that has

a finite number of possible outcomes (sample points) is called a finite sample space.

Since the events of an experiment are subsets of a universal set (the sample space

of the experiment), we may use the results for set theory given in Chapter 6 to help us

study probability. The event B is said to occur in a trial of an experiment whenever B

contains the observed outcome. We begin by explaining the roles played by the empty

set and a universal set when viewed as events associated with an experiment. The

empty set, , is called the impossible event; it cannot occur because  has no ele-

ments (outcomes). Next, the universal set S is referred to as the certain event; it must

occur because S contains all the outcomes of the experiment.

This terminology is illustrated in the next several examples.

EXAMPLE 1

Describe the sample space associated with the experiment of tossing

a coin and observing whether it falls heads or tails. What are the events of this

experiment?

Solution

The two outcomes are heads and tails, and the required sample space is

given by S  {H, T}, where H denotes the outcome heads and T denotes the outcome

tails. The events of the experiment, the subsets of S, are

, {H}, {T}, S

Note that we have included the impossible event, , and the certain event, S.

Sample Point, Sample Space, and Event

Sample point: An outcome of an experiment

Sample space: The set consisting of all possible sample points

of an experiment

Event: A subset of a sample space of an experiment

Experiment

An experiment is an activity with observable results.

7.1 Experiments, Sample Spaces, and Events

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 354

Since the events of an experiment are subsets of the sample space of the experiment,

we may talk about the union and intersection of any two events; we can also consider

the complement of an event with respect to the sample space.

Thus, the event E 傼 F contains the set of outcomes of E and/or F.

Thus, the event E 傽 F contains the set of outcomes common to E and F.

Thus, the event E

is the set containing all the outcomes in the sample space S that are

not in E.

Venn diagrams depicting the union, intersection, and complement of events are

shown in Figure 1. These concepts are illustrated in the following example.

EXAMPLE 2

Consider the experiment of rolling a die and observing the number

that falls uppermost. Let S  {1, 2, 3, 4, 5, 6} denote the sample space of the exper-

iment and E  {2, 4, 6} and F  {1, 3} be events of this experiment. Compute

(a) E 傼 F, (b) E 傽 F, and (c) F

. Interpret your results.

Solution

a. E 傼 F  {1, 2, 3, 4, 6} and is the event that the outcome of the experiment is a 1,

a 2, a 3, a 4, or a 6.

b. E 傽 F is the impossible event; the number appearing uppermost when a die

is rolled cannot be both even and odd at the same time.

c. F

 {2, 4, 5, 6} is precisely the event that the event F does not occur.

If two events cannot occur at the same time, they are said to be mutually exclu-

sive. Using set notation, we have the following definition.

Complement of an Event

The complement of event E is the event E

Intersection of Two Events

The intersection of two events E and F is the event E 傽 F.

Union of Two Events

The union of two events E and F is the event E 傼 F.

7.1 EXPERIMENTS, SAMPLE SPACES, AND EVENTS 355

FEF E

E F

(a) The union of two events (b) The intersection of two events (c) The complement of the event E

FIGURE

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 355

As before, we may use Venn diagrams to illustrate these events. In this case, the two

mutually exclusive events are depicted as two nonintersecting circles (Figure 2).

EXAMPLE 3

An experiment consists of tossing a coin three times and observing

the resulting sequence of heads and tails.

a. Describe the sample space S of the experiment.

b. Determine the event E that exactly two heads appear.

c. Determine the event F that at least one head appears.

Solution

a. The sample points may be obtained with the aid of a tree diagram (Figure 3).

The required sample space S is given by

S  {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

b. By scanning the sample space S obtained in part (a), we see that the outcomes in

which exactly two heads appear are given by the event

E  {HHT, HTH, THH}

c. Proceeding as in part (b), we find

F  {HHH, HHT, HTH, HTT, THH, THT, TTH}

EXAMPLE 4

An experiment consists of rolling a pair of dice and observing the

number that falls uppermost on each die.

a. Describe an appropriate sample space S for this experiment.

b. Determine the events E

, E

, . . . , E

that the sum of the numbers falling

uppermost is 2, 3, 4, . . . , 12, respectively.

First

toss

Sample

points

Second

toss

Third

toss

(H, H, H)

(H, H, T)

(H, T, H)

(H, T, T)

(T, H, H)

(T, H, T)

(T, T

, H)

(T, T, T)

Mutually Exclusive Events

E and F are mutually exclusive if E 傽 F .

356 7 PROBABILITY

FIGURE 2

Mutually exclusive events

FIGURE 3

Explore & Discuss

1. Suppose E and F are two

complementary events. Must

E and F be mutually exclu-

sive? Explain your answer.

2. Suppose E and F are mutu-

ally exclusive events. Must

E and F be complementary?

Explain your answer.

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 356

Solution

a. We may represent each outcome of the experiment by an ordered pair of num-

bers, the first representing the number that appears uppermost on the first die and

the second representing the number that appears uppermost on the second die. To

distinguish between the two dice, think of the first die as being red and the sec-

ond as being green. Since there are six possible outcomes for each die, the multi-

plication principle implies that there are 6



6, or 36, elements in the sample

space:

S  {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

b. With the aid of the results of part (a), we obtain the required list of events, shown

in Table 1.

APPLIED EXAMPLE 5

Movie Attendance The manager of a local

cinema records the number of patrons attending a first-run movie at the

1 p.m. screening. The theater has a seating capacity of 500.

a. What is an appropriate sample space for this experiment?

b. Describe the event E that fewer than 50 people attend the screening.

c. Describe the event F that the theater is more than half full at the

screening.

Solution

a. The number of patrons at the screening (the outcome) could run from 0 to

500. Therefore, a sample space for this experiment is

S  {0, 1, 2, 3, . . . , 500}

b. E  {0, 1, 2, 3, . . . , 49}

c. F  {251, 252, 253, . . . , 500}

7.1 EXPERIMENTS, SAMPLE SPACES, AND EVENTS 357

TABLE 1

Sum of

Uppermost Numbers Event

2 E

 {(1, 1)}

3 E

 {(1, 2), (2, 1)}

4 E

 {(1, 3), (2, 2), (3, 1)}

5 E

 {(1, 4), (2, 3), (3, 2), (4, 1)}

6 E

 {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

7 E

 {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

8 E

 {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}

9 E

 {(3, 6), (4, 5), (5, 4), (6, 3)}

10 E

 {(4, 6), (5, 5), (6, 4)}

11 E

 {(5, 6), (6, 5)}

12 E

 {(6, 6)}

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 357

APPLIED EXAMPLE 6

Family Composition An experiment consists

of studying the sex composition of a three-child family in which the

children were born at different times.

a. Describe an appropriate sample space S for this experiment.

b. Describe the event E that there are two girls and a boy in the family.

c. Describe the event F that the oldest child is a girl.

d. Describe the event G that the oldest child is a girl and the youngest child is a

boy.

Solution

a. The sample points of the experiment may be obtained with the aid of the tree

diagram shown in Figure 4, where b denotes a boy and t denotes a girl.

We see from the tree diagram that the required sample space is given by

S  {bbb, bbt, btb, btt, tbb, tbt, ttb, ttt}

Using the tree diagram, we find that:

b. E  {b

tt, tbt, ttb}

c. F  {

tbb, tbt, ttb, ttt}

d. G  {

tbb, ttb}

The next example shows that sample spaces may be infinite.

APPLIED EXAMPLE 7

Testing New Products EverBrite is develop-

ing a high-amperage, high-capacity battery as a source for powering

electric cars. The battery is tested by installing it in a prototype electric car and

running the car with a fully charged battery on a test track at a constant speed of

55 mph until the car runs out of power. The distance covered by the car is then

observed.

a. What is the sample space for this experiment?

b. Describe the event E that the driving range under test conditions is less than

150 miles.

c. Describe the event F that the driving range is between 200 and 250 miles,

inclusive.

Solution

a. Since the distance d covered by the car in any run may be given by any non-

negative number, the sample space S is given by

S  5d 0d  06

First

child

Sample

points

Second

child

Third

child

b b b

b b g

b g b

b g g

g b b

g b g

g g b

g g g

358 7 PROBABILITY

FIGURE 4

Tree diagram for three-child families

Explore & Discuss

Think of an experiment.

1. Describe the sample point(s)

and sample space of the

experiment.

2. Construct two events, E and

F, of the experiment.

3. Find the union and intersec-

tion of E and F and the com-

plement of E.

4. Are E and F mutually exclu-

sive? Explain your answer.

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 358

7.1 EXPERIMENTS, SAMPLE SPACES, AND EVENTS 359

b. The event E is given by

c. The event F is given by

F  5d 0200  d  2506

E  5d 00  d  1506

1. A sample of three apples taken from Cavallero’s Fruit

Stand is examined to determine whether the apples are

good or rotten.

a. What is an appropriate sample space for this experi-

ment?

b. Describe the event E that exactly one of the apples

picked is rotten.

c. Describe the event F that the first apple picked is rotten.

2. Refer to Self-Check Exercise 1.

a. Find E 傼 F.

b. Find E 傽 F.

c. Find F

d. Are the events E and F mutually exclusive?

Solutions to Self-Check Exercises 7.1 can be found on

page 362.

1. Explain what is meant by an experiment. Give an example.

For the example you have chosen, describe (a) a sample

point, (b) the sample space, and (c) an event of the experi-

ment.

2. What does it mean for two events to be mutually exclu-

sive? Give an example of two mutually exclusive events E

and F. How can you prove that they are mutually exclu-

sive?

In Exercises 1–6, let S ⴝ

{

a, b, c, d, e, f

}

be a sample space

of an experiment and let E ⴝ

{

a, b}, F ⴝ

{

a, d, f

}

, and G ⴝ

{

b, c, e

}

be events of this experiment.

1. Find the events E 傼 F and E 傽 F.

2. Find the events F 傼 G and F 傽 G.

3. Find the events F

and E 傽 G

4. Find the events E

and F

傽 G.

5. Are the events E and F mutually exclusive?

6. Are the events E 傼 F and E 傽 F

mutually exclusive?

In Exercises 7–14, let S ⴝ

{

1, 2, 3, 4, 5, 6

}

, E ⴝ

{

2, 4, 6

}

F ⴝ

{

1, 3, 5

}

, and G ⴝ

{

5, 6

}

7. Find the event E 傼 F 傼 G.

8. Find the event E 傽 F 傽 G.

9. Find the event (E 傼 F 傼 G)

10. Find the event (E 傽 F 傽 G)

11. Are the events E and F mutually exclusive?

12. Are the events F and G mutually exclusive?

13. Are the events E and F complementary?

14. Are the events F and G complementary?

In Exercises 15–20, let S be any sample space and let E,

F, and G be any three events associated with the experi-

ment. Describe the events using the symbols 僔, 僕, and

15. The event that E and/or F occurs

16. The event that both E and F occur

17. The event that G does not occur

18. The event that E but not F occurs

19. The event that none of the events E, F, and G occurs

20. The event that E occurs but neither of the events F or G

occurs

21. Consider the sample space S of Example 4, page 357.

a. Determine the event that the number that falls upper-

most on the first die is greater than the number that falls

uppermost on the second die.

b. Determine the event that the number that falls upper-

most on the second die is double the number that falls

uppermost on the first die.

7.1 Exercises

7.1 Self-Check Exercises

7.1 Concept Questions

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 359

22. Consider the sample space S of Example 4, page 357.

a. Determine the event that the sum of the numbers falling

uppermost is less than or equal to 7.

b. Determine the event that the number falling uppermost

on one die is a 4 and the number falling uppermost on

the other die is greater than 4.

23. Let S  {a, b, c} be a sample space of an experiment with

outcomes a, b, and c. List all the events of this experiment.

24. Let S  {1, 2, 3} be a sample space associated with an

experiment.

a. List all events of this experiment.

b. How many subsets of S contain the number 3?

c. How many subsets of S contain either the number 2 or

the number 3?

25. An experiment consists of selecting a card from a standard

deck of playing cards and noting whether it is black (B) or

red (R).

a. Describe an appropriate sample space for this experi-

ment.

b. What are the events of this experiment?

26. An experiment consists of selecting a letter at random from

the letters in the word MASSACHUSETTS and observing

the outcomes.

a. What is an appropriate sample space for this experi-

ment?

b. Describe the event “the letter selected is a vowel.”

27. An experiment consists of tossing a coin, rolling a die, and

observing the outcomes.

a. Describe an appropriate sample space for this experi-

ment.

b. Describe the event “a head is tossed and an even num-

ber is rolled.”

28. An experiment consists of spinning the hand of the num-

bered disc shown in the following figure and then observ-

ing the region in which the pointer stops. (If the needle

stops on a line, the result is discounted and the needle is

spun again.)

a. What is an appropriate sample space for this experi-

ment?

b. Describe the event “the spinner points to the number 2.”

c. Describe the event “the spinner points to an odd number.”

29. A die is rolled and the number that falls uppermost is

observed. Let E denote the event that the number shown is

a 2, and let F denote the event that the number shown is an

even number.

a. Are the events E and F mutually exclusive?

b. Are the events E and F complementary?

30. A die is rolled and the number that falls uppermost is

observed. Let E denote the event that the number shown is

even, and let F denote the event that the number is an odd

number.

a. Are the events E and F mutually exclusive?

b. Are the events E and F complementary?

31. Q

UALITY

ONTROL

A sample of three transistors taken from

a local electronics store was examined to determine whether

the transistors were defective (d) or nondefective (n). What

is an appropriate sample space for this experiment?

32. B

LOOD

YPING

Human blood is classified by the presence

or absence of three main antigens (A, B, and Rh). When a

blood specimen is typed, the presence of the A and/or B

antigen is indicated by listing the letter A and/or the letter

B. If neither the A nor B antigen is present, the letter O is

used. The presence or absence of the Rh antigen is indi-

cated by the symbols  or , respectively. Thus, if a blood

specimen is classified as AB



, it contains the A and the B

antigens as well as the Rh antigen. Similarly, O



blood

contains none of the three antigens. Using this information,

determine the sample space corresponding to the different

blood groups.

33. G

AME

HOWS

In a television game show, the winner is

asked to select three prizes from five different prizes, A, B,

C, D, and E.

a. Describe a sample space of possible outcomes (order is

not important).

b. How many points are there in the sample space corre-

sponding to a selection that includes A?

c. How many points are there in the sample space corre-

sponding to a selection that includes A and B?

d. How many points are there in the sample space corre-

sponding to a selection that includes either A or B?

34. A

UTOMATIC

ELLERS

The manager of a local bank observes

how long it takes a customer to complete his transactions

at the automatic bank teller.

a. Describe an appropriate sample space for this experi-

ment.

b. Describe the event that it takes a customer between 2

and 3 min to complete his transactions at the automatic

bank teller.

35. C

OMMON

TOCKS

Robin purchased shares of a machine

tool company and shares of an airline company. Let E be

the event that the shares of the machine tool company

increase in value over the next 6 mo, and let F be the event

that the shares of the airline company increase in value

over the next 6 mo. Using the symbols 傼, 傽, and

describe the following events.

a. The shares in the machine tool company do not increase

in value.

b. The shares in both the machine tool company and the

airline company do not increase in value.

360

7 PROBABILITY

87533_07_ch7_p353-416 1/30/08 10:04 AM Page 360

c. The shares of at least one of the two companies increase

in value.

d. The shares of only one of the two companies increase in

value.

36. C

USTOMER

ERVICE

URVEYS

The customer service depart-

ment of Universal Instruments, manufacturer of the Galaxy

home computer, conducted a survey among customers who

had returned their purchase registration cards. Purchasers

of its deluxe model home computer were asked to report

the length of time (t) in days before service was required.

a. Describe a sample space corresponding to this survey.

b. Describe the event E that a home computer required ser-

vice before a period of 90 days had elapsed.

c. Describe the event F that a home computer did not re-

quire service before a period of 1 yr had elapsed.

37. A

SSEMBLY

-T

IME

TUDIES

A time study was conducted by

the production manager of Vista Vision to determine the

length of time in minutes required by an assembly worker

to complete a certain task during the assembly of its Pulsar

color television sets.

a. Describe a sample space corresponding to this time

study.

b. Describe the event E that an assembly worker took

2 min or less to complete the task.

c. Describe the event F that an assembly worker took more

than 2 min to complete the task.

38. P

OLITICAL

OLLS

An opinion poll is conducted among a

state’s electorate to determine the relationship between

their income levels and their stands on a proposition aimed

at reducing state income taxes. Voters are classified as

belonging to either the low-, middle-, or upper-income

group. They are asked whether they favor, oppose, or are

undecided about the proposition. Let the letters L, M, and

U represent the low-, middle-, and upper-income groups,

respectively, and let the letters f, o, and u represent the

responses—favor, oppose, and undecided, respectively.

a. Describe a sample space corresponding to this poll.

b. Describe the event E

that a respondent favors the

proposition.

c. Describe the event E

that a respondent opposes the

proposition and does not belong to the low-income

group.

d. Describe the event E

that a respondent does not favor

the proposition and does not belong to the upper-income

group.

39. Q

UALITY

ONTROL

As part of a quality-control procedure,

an inspector at Bristol Farms randomly selects ten eggs

from each consignment of eggs he receives and records the

number of broken eggs.

a. What is an appropriate sample space for this experi-

ment?

b. Describe the event E that at most three eggs are broken.

c. Describe the event F that at least five eggs are broken.

40. P

OLITICAL

OLLS

In the opinion poll of Exercise 38, the

voters were also asked to indicate their political affilia-

tions—Democrat, Republican, or Independent. As before,

let the letters L, M, and U represent the low-, middle-, and

upper-income groups, respectively. Let the letters D, R,

and I represent Democrat, Republican, and Independent,

respectively.

a. Describe a sample space corresponding to this poll.

b. Describe the event E

that a respondent is a Democrat.

c. Describe the event E

that a respondent belongs to the

upper-income group and is a Republican.

d. Describe the event E

that a respondent belongs to the

middle-income group and is not a Democrat.

41. S

HUTTLE

SAGE

A certain airport hotel operates a

shuttle bus service between the hotel and the airport. The

maximum capacity of a bus is 20 passengers. On alternate

trips of the shuttle bus over a period of 1 wk, the hotel man-

ager kept a record of the number of passengers arriving at

the hotel in each bus.

a. What is an appropriate sample space for this experi-

ment?

b. Describe the event E that a shuttle bus carried fewer

than ten passengers.

c. Describe the event F that a shuttle bus arrived with a

full load.

42. S

PORTS

Eight players, A, B, C, D, E, F, G, and H, are com-

peting in a series of elimination matches of a tennis tour-

nament in which the winner of each preliminary match will

advance to the semifinals and the winners of the semifinals

will advance to the finals. An outline of the scheduled

matches follows. Describe a sample space listing the pos-

sible participants in the finals.

43. An experiment consists of selecting a card at random from

a well-shuffled 52-card deck. Let E denote the event that

an ace is drawn and let F denote the event that a spade is

drawn. Show that n(E 傼 F)  n(E)  n(F)  n(E 傽 F ).

44. Let S be a sample space for an experiment. Show that if E

is any event of an experiment, then E and E

are mutually

exclusive.

45. Let S be a sample space for an experiment, and let E and F

be events of this experiment. Show that the events E 傼 F

and E

傽 F

are mutually exclusive.

Hint: Use De Morgan’s law.

46. Let S be a sample space of an experiment with n outcomes.

Determine the number of events of this experiment.

Match I

A vs. B

C vs. D

G vs. H

E vs. F

Match II

Match III

Match IV

Semiﬁnal

match

Semiﬁnal

match

Final

match

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