Tan S.T. Finite Mathematics for the Managerial, Life, and Social Sciences
Подождите немного. Документ загружается.
1. a. Graph the histogram associated with the random vari-
able X in Example 1, page 442.
b. Find the mean and the standard deviation for these data.
2. a. Graph the histogram associated with the random vari-
able Y in Example 1, page 442.
b. Find the mean and the standard deviation for these data.
3. a. Graph the histogram associated with the data given in
Exercise 12, page 447.
b. Find the mean and the standard deviation for these data.
4. a. Graph the histogram associated with the data given in
Exercise 16, page 448.
b. Find the mean and the standard deviation for these data.
5. A sugar refiner uses a machine to pack sugar in 5-lb cartons.
To check the machine’s accuracy, cartons are selected at
random and weighed. The results follow:
4.98 5.02 4.96 4.97 5.03
4.96 4.98 5.01 5.02 5.06
4.97 5.04 5.04 5.01 4.99
4.98 5.04 5.01 5.03 5.05
4.96 4.97 5.02 5.04 4.97
5.03 5.01 5.00 5.01 4.98
a. Describe a random variable X that is associated with
these data.
b. Find the probability distribution for the random vari-
able X.
c. Compute the mean and standard deviation of X.
6. The scores of 25 students in a mathematics examination
are as follows:
90 85 74 92 68 94 66
87 85 70 72 68 73 72
69 66 58 70 74 88 90
98 71 75 68
a. Describe a random variable X that is associated with
these data.
b. Find the probability distribution for the random vari-
able X.
c. Compute the mean and standard deviation of X.
7. H
EIGHTS OF
W
OMEN
The following data, obtained from the
records of the Westwood Health Club, give the heights (to
the nearest inch) of 200 female members of the club:
Height 62 63 64 65 66
Frequency 2 3 4 8 11 20 32 30 18
Height 67 68 69 70 71
Frequency 18168105 5 4 3 2 1
a. Plot a histogram for the given data.
b. Find the mean and the standard deviation of these data.
8. A
GE
D
ISTRIBUTION IN A
T
OWN
The following table gives the
distribution of the ages (in years) of the residents (in hun-
dreds) of the town of Monroe who are under the age of 40:
Age 0–3 4–7 8–11 12–15 16–19
Residents 30 42 50 60 50
Age 20–23 24–27 28–31 32–35 36–39
Residents 41 50 45 42 34
Let X denote the random variable taking on the values 1
through 10, where 1 corresponds to the range 0–3, . . . , and
10 corresponds to the range 36–39.
a. Plot a histogram for the given data.
b. Find the mean and the standard deviation of X.
70
1
2
69
1
2
68
1
2
67
1
2
66
1
2
65
1
2
64
1
2
63
1
2
62
1
2
452 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
TECHNOLOGY EXERCISES
Bernoulli Trials
An important class of experiments have (or may be viewed as having) two outcomes.
For example, in a coin-tossing experiment, the two outcomes are heads and tails. In
the card game played by Mike and Bill (Example 7, Section 8.2), one may view the
selection of a diamond as a win (for Mike) and the selection of a card of another suit
as a loss for Mike. For a third example, consider an experiment in which a person is
inoculated with a flu vaccine. Here, the vaccine may be classified as being “effective”
or “ineffective” with respect to that particular person.
8.4 The Binomial Distribution
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 452
In general, experiments with two outcomes are called Bernoulli trials, or
binomial trials. It is standard practice to label one of the outcomes of a binomial
trial a success and the other a failure. For example, in a coin-tossing experiment,
the outcome a head may be called a success, in which case the outcome a tail is
called a failure. Note that by using the terms success and failure in this way, we
depart from their usual connotations.
A sequence of Bernoulli (binomial) trials is called a binomial experiment. More
precisely, we have the following definition:
In a binomial experiment, it is customary to denote the number of trials by n, the
probability of a success by p, and the probability of a failure by q. Because the event of
a success and the event of a failure are complementary events, we have the relationship
p q 1
or, equivalently,
q 1 p
The following example involves a binomial experiment.
EXAMPLE 1
A fair die is rolled four times. Compute the probability of obtaining
exactly one 6 in the four throws.
Solution
There are four trials in this experiment. Each trial consists of rolling the
die once and observing the face that lands uppermost. We may view each trial as an
experiment with two outcomes: a success (S) if the face that lands uppermost is a 6
and a failure (F) if it is any of the other five numbers. Letting p and q denote the
probability of success and failure, respectively, of a single trial of the experiment,
we find that
Furthermore, we may assume that the trials of this experiment are independent.
Thus, we have a binomial experiment.
With the aid of the multiplication principle, we see that the experiment has 2
4
,
or 16, outcomes. We can obtain these outcomes by constructing the tree diagram
associated with the experiment (see Table 11, where the outcomes are listed accord-
ing to the number of successes). From the table, we see that the event of obtaining
exactly one success in four trials is given by
E {SFFF, FSFF, FFSF, FFFS}
with probability given by
P(E) P(SFFF ) P(FSFF ) P(FFSF ) P(FFFS) (10)
p
1
6
and
q 1
1
6
5
6
Binomial Experiment
A binomial experiment has the following properties:
1. The number of trials in the experiment is fixed.
2. There are two outcomes of each trial: “success” and “failure.”
3. The probability of success in each trial is the same.
4. The trials are independent of each other.
8.4 THE BINOMIAL DISTRIBUTION 453
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 453
454 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
Since the trials (throws) are independent, the terms on the right-hand side of Equa-
tion (10) may be computed as follows:
P(SFFF) P(S)P(F)P(F)P(F) p
q
q
q pq
3
P(FSFF) P(F)P(S)P(F)P(F) q
p
q
q pq
3
P(FFSF) P(F)P(F)P(S)P(F) q
q
p
q pq
3
P(FFFS) P(F)P(F)P(F)P(S) q
q
q
p pq
3
Therefore, upon substituting these values in (10), we obtain
Probabilities in Bernoulli Trials
Let’s reexamine the computations performed in the last example. There it was found
that the probability of obtaining exactly one success in a binomial experiment with
four independent trials with probability of success in a single trial p is given by
P(E) 4pq
3
(where q 1 p) (11)
Observe that the coefficient 4 of pq
3
appearing in Equation (11) is precisely the num-
ber of outcomes of the experiment with exactly one success and three failures, the out-
comes being
SFFF FSFF FFSF FFFS
Another way of obtaining this coefficient is to think of the outcomes as arrangements
of the letters S and F. Then, the number of ways of selecting one position for S from
four possibilities is given by
Next, observe that, because the trials are independent, each of the four outcomes of
the experiment has the same probability, given by
pq
3
where the exponents 1 and 3 of p and q, respectively, correspond to exactly one suc-
cess and three failures in the trials that make up each outcome.
As a result of the foregoing discussion, we may write (11) as
P(E) C(4, 1)pq
3
(12)
4
C
14, 12
4!
1! 14 12!
4 a
1
6
b a
5
6
b
3
⬇ .386
P 1E 2 pq
3
pq
3
pq
3
pq
3
4pq
3
TABLE 11
01 2 3 4
Success Success Successes Successes Successes
FFFF SFFF SSFF SSSF SSSS
FSFF SFSF SSFS
FFSF SFFS SFSS
FFFS FSSF FSSS
FSFS
FFSS
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 454
We are also in a position to generalize this result. Suppose that in a binomial experi-
ment the probability of success in any trial is p. What is the probability of obtaining
exactly x successes in n independent trials? We start by counting the number of out-
comes of the experiment, each of which has exactly x successes. Now, one such out-
come involves x successive successes followed by (n x) failures—that is,
SS
. . .
S FF
. . .
F (13)
xn x
The other outcomes, each of which has exactly x successes, are obtained by re-
arranging the S’s (x of them) and F’s (n x of them). There are C(n, x) ways of ar-
ranging these letters. Next, arguing as in Example 1, we see that each such outcome
has probability given by
p
x
q
nx
For example, for the outcome (13) we find
P(SS
. . .
S FF
. . .
F) P(S)P(S)
. . .
P(S) P(F)P(F)
. . .
P(F)
x (n x) x (n x)
pp
. . .
p qq
. . .
q
xn x
p
x
q
nx
Let’s now state this important result formally.
If we let X be the random variable that gives the number of successes in a binomial
experiment, then the probability of exactly x successes in n independent trials may be
written
P(X x) C(n, x)p
x
q
nx
(x 0, 1, 2, . . . , n) (14)
The random variable X is called a binomial random variable, and the probability dis-
tribution of X is called a binomial distribution.
EXAMPLE 2
A fair die is rolled five times. If a 1 or a 6 lands uppermost in a trial,
then the throw is considered a success. Otherwise, the throw is considered a failure.
a. Find the probability of obtaining exactly 0, 1, 2, 3, 4, and 5 successes, respec-
tively, in this experiment.
b. Using the results obtained in the solution to part (a), construct the binomial distri-
bution for this experiment and draw the histogram associated with it.
Solution
a. This is a binomial experiment with X, the binomial random variable, taking on
each of the values 0, 1, 2, 3, 4, and 5 corresponding to exactly 0, 1, 2, 3, 4, and
5 successes, respectively, in five trials. Since the die is fair, the probability of a
1 or a 6 landing uppermost in any trial is given by p , from which it also
follows that q 1 p . Finally, n 5 since there are five trials (throws of
2
3
1
3
2
6
Computation of Probabilities in Bernoulli Trials
In a binomial experiment in which the probability of success in any trial is p, the
probability of exactly x successes in n independent trials is given by
C(n, x)p
x
q
nx
8.4 THE BINOMIAL DISTRIBUTION 455
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 455
the die) in this experiment. Using Equation (14), we find that the required proba-
bilities are
b. Using these results, we find the required binomial distribution associated with
this experiment given in Table 12. Next, we use this table to construct the
histogram associated with the probability distribution (Figure 12).
EXAMPLE 3
A fair die is rolled five times. If a 1 or a 6 lands uppermost in a trial,
then the throw is considered a success. Use the results from Example 2 to answer
the following questions:
a. What is the probability of obtaining 0 or 1 success in the experiment?
b. What is the probability of obtaining at least 1 success in the experiment?
Solution
Using Table 12, we find that
a. The probability of obtaining 0 or 1 success in the experiment is given by
P(X 0) P(X 1) .132 .329 .461
b. The probability of obtaining at least 1 success in the experiment is given by
P(X 1) P(X 2) P(X 3) P(X 4) P(X 5)
.329 .329 .165 .041 .004
.868
This result may also be obtained by observing that 1 P(X 0) 1 .132 .868.
The following formulas (which we state without proof ) will be useful in solving
problems that involve binomial experiments.
P 1X 5 2 C 15, 5 2 a
1
3
b
5
a
2
3
b
0
5!
5! 0!
#
1
243
⬇ .004
P 1X 4 2 C 15, 4 2 a
1
3
b
4
a
2
3
b
1
5!
4! 1!
#
2
243
⬇ .041
P 1X 3 2 C 15, 3 2 a
1
3
b
3
a
2
3
b
2
5!
3! 2!
#
4
243
⬇ .165
P 1X 2 2 C 15, 2 2 a
1
3
b
2
a
2
3
b
3
5!
2! 3!
#
8
243
⬇ .329
P 1X 1 2 C 15, 1 2 a
1
3
b
1
a
2
3
b
4
5!
1! 4!
#
16
243
⬇ .329
P 1X 0 2 C 15, 0 2 a
1
3
b
0
a
2
3
b
5
5!
0! 5!
#
32
243
⬇ .132
456 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
012345
.4
.3
.2
.1
x
FIGURE 12
The probability of the number of suc-
cesses in five throws
TABLE 12
Probability Distribution
xP(X x)
0 .132
1 .329
2 .329
3 .165
4 .041
5 .004
Explore & Discuss
Consider the equation
P(X x) C(n, x)p
x
q
nx
for the binomial distribution.
1. Construct the histogram with
n 5 and p .2; the histo-
gram with n 5 and p .5;
and the histogram with n 5
and p .8.
2. Comment on the shape of the
histograms and give an inter-
pretation.
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 456
EXAMPLE 4
For the experiment in Examples 2 and 3, compute the mean, the
variance, and the standard deviation of X by (a) using Formulas (15a), (15b), and
(15c) and (b) using the definition of each term (Sections 8.2 and 8.3).
Solution
a. We use (15a), (15b), and (15c), with , and n ⫽ 5, obtaining
We leave it to you to interpret the results.
b. Using the definition of expected value (Section 8.2) and the values of the proba-
bility distribution shown in Table 12, we find that
⫽ E(X) ⬇ (0)(.132) ⫹ (1)(.329) ⫹ (2)(.329)
⫹ (3)(.165) ⫹ (4)(.041) ⫹ (5)(.004)
⬇ 1.67
which agrees with the result obtained in part (a). Next, using the definition of
variance and ⫽ 1.67, we find that
which again agrees with the preceding results.
We close this section by looking at several examples involving binomial experiments.
In working through these examples, you may use a calculator or you may consult
Table 1, Binomial Probabilities, Appendix D.
APPLIED EXAMPLE 5
Quality Control A division of Solaron manu-
factures photovoltaic cells to use in the company’s solar energy converters.
It is estimated that 5% of the cells manufactured are defective. If a random sam-
ple of 20 is selected from a large lot of cells manufactured by the company, what
is the probability that it will contain at most 2 defective cells?
Solution
We may view this as a binomial experiment. To see this, first note that
a fixed number of trials (n ⫽ 20) corresponds to the selection of exactly 20 photo-
voltaic cells. Second, observe that there are exactly two outcomes in the experi-
ment, defective (“success”) and nondefective (“failure”). Third, the probability of
⬇ 11.11 ⬇ 1.05
s
X
⫽ 1Var 1X2
⬇ 1.11
⫹ 1.1652 11.332
2
⫹ 1.0412 12.332
2
⫹ 1.0042 13.332
2
Var 1X2⫽ 1.1322 1⫺1.672
2
⫹ 1.3292 1⫺0.672
2
⫹ 1.3292 10.332
2
s
X
⫽ 1Var 1X2⬇ 11.11 ⬇ 1.05
Var
1X2⫽ 152a
1
3
ba
2
3
b⫽
10
9
⬇ 1.11
m ⫽ E 1X 2⫽ 152a
1
3
b⫽
5
3
⬇ 1.67
p ⫽
1
3
, q ⫽
2
3
8.4 THE BINOMIAL DISTRIBUTION 457
Mean, Variance, and Standard Deviation of a Random Variable X
If X is a binomial random variable associated with a binomial experiment con-
sisting of n trials with probability of success p and probability of failure q, then
the mean (expected value), variance, and standard deviation of X are
(15a)
(15b)
(15c) s
X
⫽ 1npq
Var 1X2⫽ npq
m ⫽ E 1X 2⫽ np
87533_08_ch8_p417-482 2/18/08 10:45 AM Page 457
success in each trial is .05 ( p .05) and the probability of failure in each trial is
.95 (q .95). This assumption is justified by virtue of the fact that the lot from
which the cells are selected is “large,” so the removal of a few cells will not appre-
ciably affect the percentage of defective cells in the lot in each successive trial.
Finally, the trials are independent of each other—once again because of the lot size.
Letting X denote the number of defective cells, we find that the probability of
finding at most 2 defective cells in the sample of 20 is given by
P(X 0) P(X 1) P(X 2)
C(20, 0)(.05)
0
(.95)
20
C(20, 1)(.05)
1
(.95)
19
C(20, 2)(.05)
2
(.95)
18
⬇ .3585 .3774 .1887
.9246
Thus, for lots of photovoltaic cells manufactured by Solaron, approximately 92%
of the samples will have at most 2 defective cells; equivalently, approximately
8% of the samples will contain more than 2 defective cells.
APPLIED EXAMPLE 6
Success of Heart Transplants The probabil-
ity that a heart transplant performed at the Medical Center is successful
(that is, the patient survives 1 year or more after undergoing such an operation) is
.7. Of six patients who have recently undergone such an operation, what is the
probability that, 1 year from now,
a. None of the heart recipients will be alive?
b. Exactly three will be alive?
c. At least three will be alive?
d. All will be alive?
Solution
Here, n 6, p .7, and q .3. Let X denote the number of success-
ful operations. Then:
a. The probability that no heart recipients will be alive after 1 year is given by
b. The probability that exactly three will be alive after 1 year is given by
c. The probability that at least three will be alive after 1 year is given by
.9294
⬇ .1852 .3241 .3025 .1176
6!
6! 0!
1.72
6
#
1
6!
3! 3!
1.72
3
1.32
3
6!
4! 2!
1.72
4
1.32
2
6!
5! 1!
1.72
5
1.32
1
C 16, 52 1.72
5
1.32
1
C 16, 62 1.72
6
1.32
0
C 16, 32 1.72
3
1.32
3
C 16, 42 1.72
4
1.32
2
P 1X 32 P 1X 42 P1X 52 P1X 6 2
⬇ .1852
6!
3! 3!
1.72
3
1.32
3
P 1X 3 2 C 16, 3 2 1.7 2
3
1.32
3
⬇ .0007
6!
0! 6!
#
1
#
1.32
6
P 1X 0 2 C 16, 0 2 1.7 2
0
1.32
6
458 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 458
1. A binomial experiment consists of four independent trials.
The probability of success in each trial is .2.
a. Find the probability of obtaining exactly 0, 1, 2, 3, and
4 successes, respectively, in this experiment.
b. Construct the binomial distribution and draw the his-
togram associated with this experiment.
c. Compute the mean and the standard deviation of the
random variable associated with this experiment.
2. A survey shows that 60% of the households in a large met-
ropolitan area have microwave ovens. If ten households are
selected at random, what is the probability that five or
fewer of these households have microwave ovens?
Solutions to Self-Check Exercises 8.4 can be found on
page 462.
d. The probability that all will be alive after 1 year is given by
APPLIED EXAMPLE 7
Quality Control PAR Bearings manufactures
ball bearings packaged in lots of 100 each. The company’s quality-control
department has determined that 2% of the ball bearings manufactured do not meet
the specifications imposed by a buyer. Find the average number of ball bearings
per package that fail to meet the buyer’s specification.
Solution
The experiment under consideration is binomial. The average number
of ball bearings per package that fail to meet with the specifications is therefore
given by the expected value of the associated binomial random variable. Using
(15a), we find that
E(X ) np (100)(.02) 2
substandard ball bearings in a package of 100.
⬇ .1176
P 1X 6 2 C 16, 6 2 1.7 2
6
1.32
0
6!
6! 0!
1.72
6
#
1
8.4 THE BINOMIAL DISTRIBUTION 459
8.4 Exercises
8.4 Self-Check Exercises
8.4 Concept Questions
1. Suppose that you are given a Bernoulli experiment.
a. How many outcomes are there in each trial?
b. Can the number of trials in the experiment vary, or is it
fixed?
c. Are the trials in the experiment dependent?
d. If the probability of success in any trial is p, what is
the probability of exactly x successes in n independent
trials?
2. Give the formula for the mean, variance, and standard devi-
ation of X, where X is a binomial random variable associ-
ated with a binomial experiment consisting of n trials with
probability of success p and probability of failure q.
In Exercises 1–6, determine whether the experiment is a
binomial experiment. Justify your answer.
1. Rolling a fair die three times and observing the number of
times a 6 is thrown
2. Rolling a fair die and observing the number of times the die
is thrown until a 6 appears uppermost
3. Rolling a fair die three times and observing the number that
appears uppermost
4. A card is selected from a deck of 52 cards, and its color is
observed. A second card is then drawn (without replace-
ment), and its color is observed.
5. Recording the number of accidents that occur at a given
intersection on four clear days and one rainy day
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 459
6. Recording the number of hits a baseball player, whose bat-
ting average is .325, gets after being up to bat five times
In Exercises 7–10, find C(n, x)p
x
q
nx
for the given values
of n, x, and p.
7. 8.
9. n 5, x 3, p .2 10. n 6, x 5, p .4
In Exercises 11–16, use the formula C(n, x)p
x
q
nx
to
determine the probability of the given event.
11. The probability of exactly no successes in five trials of a
binomial experiment in which
12. The probability of exactly three successes in six trials of a
binomial experiment in which
13. The probability of at least three successes in six trials of a
binomial experiment in which
14. The probability of no successful outcomes in six trials of a
binomial experiment in which
15. The probability of no failures in five trials of a binomial
experiment in which
16. The probability of at least one failure in five trials of a
binomial experiment in which
17. A fair die is rolled four times. Calculate the probability of
obtaining exactly two 6s.
18. Let X be the number of successes in five independent trials
in a binomial experiment in which the probability of suc-
cess is Find:
a. P(X 4) b. P(2 X 4)
19. A binomial experiment consists of five independent trials.
The probability of success in each trial is .4.
a. Find the probability of obtaining exactly 0, 1, 2, 3, 4,
and 5 successes, respectively, in this experiment.
b. Construct the binomial distribution and draw the his-
togram associated with this experiment.
c. Compute the mean and the standard deviation of the
random variable associated with this experiment.
20. Let the random variable X denote the number of girls in a
five-child family. If the probability of a female birth is .5,
a. Find the probability of 0, 1, 2, 3, 4, and 5 girls in a five-
child family.
b. Construct the binomial distribution and draw the his-
togram associated with this experiment.
c. Compute the mean and the standard deviation of the
random variable X.
21. The probability that a fuse produced by a certain manufac-
turing process will be defective is . Is it correct to infer
from this statement that there is at most 1 defective fuse
in each lot of 50 produced by this process? Justify your
answer.
1
50
p
2
5
.
p
1
3
p
1
3
p
1
3
p
1
2
p
1
2
p
1
3
n 6, x 4, p
1
4
n 4, x 2, p
1
3
22. S
PORTS
If the probability that a certain tennis player will
serve an ace is , what is the probability that he will serve
exactly two aces out of five serves? (Assume that the five
serves are independent.)
23. S
PORTS
If the probability that a certain tennis player will
serve an ace is .15, what is the probability that she will
serve at least two aces out of five serves? (Assume that the
five serves are independent.)
24. S
ALES
P
REDICTIONS
From experience, the manager of
Kramer’s Book Mart knows that 40% of the people who
are browsing in the store will make a purchase. What is the
probability that, among ten people who are browsing in the
store, at least three will make a purchase?
25. C
USTOMER
S
ERVICES
Mayco, a mail-order department store,
has six telephone lines available for customers who wish to
place their orders. If the probability is that any one of the
six telephone lines is engaged during business hours, find
the probability that all six lines will be in use when a cus-
tomer calls to place an order.
26. R
ESTAURANT
V
IOLATIONS OF THE
H
EALTH
C
ODE
Suppose
30% of the restaurants in a certain part of a town are in vio-
lation of the health code. If a health inspector randomly
selects five of the restaurants for inspection, what is the
probability that
a. None of the restaurants are in violation of the health
code?
b. One of the restaurants is in violation of the health code?
c. At least two of the restaurants are in violation of the
health code?
27. A
DVERTISEMENTS
An advertisement for brand A chicken
noodle soup claims that 60% of all consumers prefer brand
A over brand B, the chief competitor’s product. To test this
claim, David Horowitz, host of The Consumer Advocate,
selected ten people at random. After tasting both soups,
each person was asked to state his or her preference.
Assuming the company’s claim is correct, find the proba-
bility that
a. Six or more people stated a preference for brand A.
b. Fewer than six people stated a preference for brand A.
28. V
OTERS
In a certain congressional district, it is known that
40% of the registered voters classify themselves as conser-
vatives. If ten registered voters are selected at random from
this district, what is the probability that four of them will
be conservatives?
29. V
IOLATIONS OF THE
B
UILDING
C
ODE
Suppose that one third
of the new buildings in a town are in violation of the build-
ing code. If a building inspector inspects five of the build-
ings chosen at random, find the probability that
a. The first three buildings will pass the inspection and the
remaining two will fail the inspection.
b. Exactly three of the buildings will pass inspection.
1
4
1
4
460 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 460
30. E
XAMS
A biology quiz consists of eight multiple-choice
questions. Five must be answered correctly to receive a
passing grade. If each question has five possible answers,
of which only one is correct, what is the probability that a
student who guesses at random on each question will pass
the examination?
31. B
LOOD
T
YPES
It is estimated that one third of the general
population has blood type A
. If a sample of nine people
is selected at random, what is the probability that
a. Exactly three of them have blood type A
?
b. At most three of them have blood type A
?
32. E
XAMS
A psychology quiz consists of ten true-or-false
questions. If a student knows the correct answer to six of
the questions but determines the answers to the remaining
questions by flipping a coin, what is the probability that
she will obtain a score of at least 90%?
33. Q
UALITY
C
ONTROL
The probability that a DVD player pro-
duced by VCA Television is defective is estimated to be
.02. If a sample of ten players is selected at random, what
is the probability that the sample contains
a. No defectives? b. At most two defectives?
34. Q
UALITY
C
ONTROL
As part of its quality-control program,
the video-game DVDs produced by Starr Communications
are subjected to a final inspection before shipment. A sam-
ple of six DVDs is selected at random from each lot of
DVDs produced, and the lot is rejected if the sample con-
tains one or more defective DVDs. If 1.5% of the DVDs
produced by Starr is defective, find the probability that a
shipment will be accepted.
35. R
OBOT
R
ELIABILITY
An automobile-manufacturing com-
pany uses ten industrial robots as welders on its assembly
line. On a given working day, the probability that a robot
will be inoperative is .05. What is the probability that, on a
given working day,
a. Exactly two robots are inoperative?
b. More than two robots are inoperative?
36. E
NGINE
F
AILURES
The probability that an airplane engine
will fail in a transcontinental flight is .001. Assuming that
engine failures are independent of each other, what is the
probability that, on a certain transcontinental flight, a four-
engine plane will experience
a. Exactly one engine failure?
b. Exactly two engine failures?
c. More than two engine failures? (Note: In this event, the
airplane will crash.)
37. Q
UALITY
C
ONTROL
The manager of Toy World has decided
to accept a shipment of electronic games if none of a ran-
dom sample of 20 is found to be defective. What is the
probability that he will accept the shipment
a. If 10% of the electronic games is defective?
b. If 5% of the electronic games is defective?
38. Q
UALITY
C
ONTROL
Refer to Exercise 37. If the manager’s
criterion for accepting shipment is that there be no more
than 1 defective electronic game in a random sample of 20,
what is the probability that he will accept the shipment if
10% of the electronic games is defective?
39. Q
UALITY
C
ONTROL
Refer to Exercise 37. If the manager of
the store changes his sample size to 10 and decides to
accept shipment only if none of the sampled games is
defective, what is the probability that he will accept the
shipment if 10% of the games is defective?
40. M
AIL
-O
RDER
S
ALES
Ruth, the owner of a mail-order house,
estimates that the probability that a household receiving
one of her catalogs will place an order with her is .10. How
many catalogs must Ruth send out to ensure that the
chances of obtaining at least one order is 50% or better?
41. How many times must a person toss a fair coin to ensure
that the chances of obtaining at least one head to be 99% or
better?
42. F
LIPPING A
C
OIN
An unbiased coin is tossed 1 million times.
Show that the probability is at least .99 that the proportion
of times the coin will land heads is between .495 and .505,
inclusive. (Note: These results show that if an unbiased coin
is tossed a very large number times, the proportion of times
that the coin will land heads is very close to .)
Hint: Use Chebychev’s inequality.
43. D
RUG
T
ESTING
A new drug has been found to be effective
in treating 75% of the people afflicted by a certain disease.
If the drug is administered to 500 people who have this dis-
ease, what are the mean and the standard deviation of the
number of people for whom the drug can be expected to be
effective?
44. C
OLLEGE
G
RADUATES
At a certain university, the probabil-
ity that an entering freshman will graduate within 4 yr is .6.
From an incoming class of 2000 freshmen, find
a. The expected number of students who will graduate
within 4 yr.
b. The standard deviation of the number of students who
will graduate within 4 yr.
In Exercises 45–48, 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.
45. In a binomial trial, the number of outcomes of the experi-
ment may be any finite number.
46. In a binomial experiment with n 3, P(X 1 or 2) 3pq.
47. If the probability that a batter gets a hit is at each time at
bat, then the batter is sure to get a hit if she bats four times.
48. The histogram associated with a binomial distribution is
symmetric with respect to if .p
1
2
x
n
2
1
4
1
2
8.4 THE BINOMIAL DISTRIBUTION 461
87533_08_ch8_p417-482 1/30/08 10:09 AM Page 461