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5.4 Binomial Probability Distribution 215
For the Martin Clothing Store problem with three customers, we can use equation (5.9)
to compute the expected number of customers who will make a purchase.
Suppose that for the next month the Martin Clothing Store forecasts that 1000 cus-
tomers will enter the store. What is the expected number of customers who will make a pur-
chase? The answer is μ ⫽ np ⫽ (1000)(.3) ⫽ 300. Thus, to increase the expected number
of purchases, Martin’s must induce more customers to enter the store and/or somehow in-
crease the probability that any individual customer will make a purchase after entering.
For the Martin Clothing Store problem with three customers, we see that the variance
and standard deviation for the number of customers who will make a purchase are
For the next 1000 customers entering the store, the variance and standard deviation for
the number of customers who will make a purchase are
σ
2
⫽
σ ⫽
np(1 ⫺ p) ⫽ 1000
(.3)(.7) ⫽ 210
兹
210 ⫽ 14.49
σ
2
⫽
σ ⫽
np(1 ⫺ p) ⫽ 3(.3)(.7) ⫽ .63
兹
.63 ⫽ .79
E(x) ⫽ np ⫽ 3(.30) ⫽ .9
NOTES AND COMMENTS
1. The binomial table in Appendix B shows values
of p up to and including p ⫽ .95. Some sources
of the binomial table only show values of p up to
and including p ⫽ .50. It would appear that such
a table cannot be used when the probability of
success exceeds p ⫽ .50. However, the table can
be used by noting that the probability of n ⫺ x
failures is also the probability of x successes.
Thus, when the probability of success is greater
than p ⫽ .50, we can compute the probability of
n ⫺ xfailures instead. The probability of failure,
1 ⫺ p, will be less than .50 when p ⬎ .50.
2. Some sources present the binomial table in a cu-
mulative form. In using such a table, one must
subtract entries in the table to find the probabil-
ity of exactly x success in n trials. For example,
f(2) ⫽ P(x ⱕ 2) ⫺ P(x ⱕ 1). The binomial table
we provide in Appendix B provides f(2) directly.
To compute cumulative probabilities using the
binomial table in Appendix B, sum the entries in
the table. For example, to determine the cumu-
lative probability P(x ⱕ 2), compute the sum
f(0) ⫹ f(1) ⫹ f(2).
x P(X = x)
0 0.0282
1 0.1211
2 0.2335
3 0.2668
4 0.2001
5 0.1029
6 0.0368
7 0.0090
8 0.0014
9 0.0001
10 0.0000
FIGURE 5.5 MINITAB OUTPUT SHOWING BINOMIAL PROBABILITIES
FOR THE MARTIN CLOTHING STORE PROBLEM
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216 Chapter 5 Discrete Probability Distributions
Exercises
Methods
25. Consider a binomial experiment with two trials and p ⫽ .4.
a. Draw a tree diagram for this experiment (see Figure 5.3).
b. Compute the probability of one success, f(1).
c. Compute f(0).
d. Compute f(2).
e. Compute the probability of at least one success.
f. Compute the expected value, variance, and standard deviation.
26. Consider a binomial experiment with n ⫽ 10 and p ⫽ .10.
a. Compute f(0).
b. Compute f(2).
c. Compute P(x ⱕ 2).
d. Compute P(x ⱖ 1).
e. Compute E(x).
f. Compute Var(x) and σ.
27. Consider a binomial experiment with n ⫽ 20 and p ⫽ .70.
a. Compute f(12).
b. Compute f(16).
c. Compute P(x ⱖ 16).
d. Compute P(x ⱕ 15).
e. Compute E(x).
f. Compute Var(x) and σ.
Applications
28. AHarris Interactive survey for InterContinental Hotels & Resorts asked respondents, “When
traveling internationally, do you generally venture out on your own to experience culture,
or stick with your tour group and itineraries?” The survey found that 23% of the respon-
dents stick with their tour group (USA Today, January 21, 2004).
a. In a sample of six international travelers, what is the probability that two will stick
with their tour group?
b. In a sample of six international travelers, what is the probability that at least two will
stick with their tour group?
c. In a sample of 10 international travelers, what is the probability that none will stick
with the tour group?
29. In San Francisco, 30% of workers take public transportation daily (USA Today, December
21, 2005).
a. In a sample of 10 workers, what is the probability that exactly 3 workers take public
transportation daily?
b. In a sample of 10 workers, what is the probability that at least 3 workers take public
transportation daily?
30. When a new machine is functioning properly, only 3% of the items produced are defective.
Assume that we will randomly select two parts produced on the machine and that we are
interested in the number of defective parts found.
a. Describe the conditions under which this situation would be a binomial experiment.
b. Draw a tree diagram similar to Figure 5.3 showing this problem as a two-trial experiment.
c. How many experimental outcomes result in exactly one defect being found?
d. Compute the probabilities associated with finding no defects, exactly one defect, and
two defects.
test
SELF
test
SELF
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5.4 Binomial Probability Distribution 217
31. A Randstad/Harris interactive survey reported that 25% of employees said their company
is loyal to them (USAToday, November 11, 2009). Suppose 10 employees are selected ran-
domly and will be interviewed about company loyalty.
a. Is the selection of 10 employees a binomial experiment? Explain.
b. What is the probability that none of the 10 employees will say their company is loyal
to them?
c. What is the probability that 4 of the 10 employees will say their company is loyal to them?
d. What is the probability that at least 2 of the 10 employees will say their company is
loyal to them?
32. Military radar and missile detection systems are designed to warn a country of an enemy at-
tack. Areliability question is whether a detection system will be able to identify an attack and
issue a warning. Assume that a particular detection system has a .90 probability of detecting
a missile attack. Use the binomial probability distribution to answer the following questions.
a. What is the probability that a single detection system will detect an attack?
b. If two detection systems are installed in the same area and operate independently, what
is the probability that at least one of the systems will detect the attack?
c. If three systems are installed, what is the probability that at least one of the systems
will detect the attack?
d. Would you recommend that multiple detection systems be used? Explain.
33. Twelve of the top 20 finishers in the 2009 PGA Championship at Hazeltine National Golf
Club in Chaska, Minnesota, used a Titleist brand golf ball (GolfBallTest website,
November 12, 2009). Suppose these results are representative of the probability that a ran-
domly selected PGA Tour player uses a Titleist brand golf ball. For a sample of 15 PGA
Tour players, make the following calculations.
a. Compute the probability that exactly 10 of the 15 PGATour players use a Titleist brand
golf ball.
b. Compute the probability that more than 10 of the 15 PGA Tour players use a Titleist
brand golf ball.
c. For a sample of 15 PGA Tour players, compute the expected number of players who
use a Titleist brand golf ball.
d. For a sample of 15 PGATour players, compute the variance and standard deviation of
the number of players who use a Titleist brand golf ball.
34. The Census Bureau’s Current Population Survey shows that 28% of individuals, ages 25 and
older, have completed four years of college (The New York Times Almanac, 2006). For a sam-
ple of 15 individuals, ages 25 and older, answer the following questions:
a. What is the probability that four will have completed four years of college?
b. What is the probability that three or more will have completed four years of college?
35. Auniversity found that 20% of its students withdraw without completing the introductory
statistics course. Assume that 20 students registered for the course.
a. Compute the probability that two or fewer will withdraw.
b. Compute the probability that exactly four will withdraw.
c. Compute the probability that more than three will withdraw.
d. Compute the expected number of withdrawals.
36. According to a survey conducted by TD Ameritrade, one out of four investors have
exchange-traded funds in their portfolios (USA Today, January 11, 2007). Consider a sample
of 20 investors.
a. Compute the probability that exactly four investors have exchange-traded funds in
their portfolios.
b. Compute the probability that at least two of the investors have exchange-traded funds
in their portfolios.
c. If you found that exactly 12 of the investors have exchange-traded funds in their port-
folios, would you doubt the accuracy of the survey results?
d. Compute the expected number of investors who have exchange-traded funds in their
portfolios.
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218 Chapter 5 Discrete Probability Distributions
37. Twenty-three percent of automobiles are not covered by insurance (CNN, February 23,
2006). On a particular weekend, 35 automobiles are involved in traffic accidents.
a. What is the expected number of these automobiles that are not covered by insurance?
b. What are the variance and standard deviation?
The Poisson probability
distribution is often used to
model random arrivals in
waiting line situations.
PROPERTIES OF A POISSON EXPERIMENT
1. The probability of an occurrence is the same for any two intervals of equal length.
2. The occurrence or nonoccurrence in any interval is independent of the occur-
rence or nonoccurrence in any other interval.
POISSON PROBABILITY FUNCTION
(5.11)
where
f(x) ⫽
μ ⫽
e ⫽
the probability of x occurrences in an interval
expected value or mean number of occurrences
2.71828
in an interval
f(x) ⫽
μ
x
e
⫺μ
x!
Siméon Poisson taught
mathematics at the École
Polytechnique in Paris
from 1802 to 1808. In 1837,
he published a work
entitled, “Researches on
the Probability of Criminal
and Civil Verdicts,” which
includes a discussion of
what later became known
as the Poisson distribution.
The Poisson probability function is defined by equation (5.11).
For the Poisson probability distribution, x is a discrete random variable indicating the num-
ber of occurrences in the interval. Since there is no stated upper limit for the number of oc-
currences, the probability function f(x) is applicable for values x ⫽ 0, 1, 2,...without
limit. In practical applications, x will eventually become large enough so that f(x) is ap-
proximately zero and the probability of any larger values of x becomes negligible.
An Example Involving Time Intervals
Suppose that we are interested in the number of arrivals at the drive-up teller window of a
bank during a 15-minute period on weekday mornings. If we can assume that the probability
of a car arriving is the same for any two time periods of equal length and that the arrival or
nonarrival of a car in any time period is independent of the arrival or nonarrival in any other
time period, the Poisson probability function is applicable. Suppose these assumptions are
satisfied and an analysis of historical data shows that the average number of cars arriving in
a 15-minute period of time is 10; in this case, the following probability function applies.
f(x) ⫽
10
x
e
⫺10
x!
Bell Labs used the Poisson
distribution to model the
arrival of telephone calls.
5.5 Poisson Probability Distribution
In this section we consider a discrete random variable that is often useful in estimating the
number of occurrences over a specified interval of time or space. For example, the random
variable of interest might be the number of arrivals at a car wash in one hour, the number
of repairs needed in 10 miles of highway, or the number of leaks in 100 miles of pipeline.
If the following two properties are satisfied, the number of occurrences is a random vari-
able described by the Poisson probability distribution.
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5.5 Poisson Probability Distribution 219
μ
x 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10
0 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0000
1 .0010 .0009 .0009 .0008 .0007 .0007 .0006 .0005 .0005 .0005
2 .0046 .0043 .0040 .0037 .0034 .0031 .0029 .0027 .0025 .0023
3 .0140 .0131 .0123 .0115 .0107 .0100 .0093 .0087 .0081 .0076
4 .0319 .0302 .0285 .0269 .0254 .0240 .0226 .0213 .0201 .0189
5 .0581 .0555 .0530 .0506 .0483 .0460 .0439 .0418 .0398 .0378
6 .0881 .0851 .0822 .0793 .0764 .0736 .0709 .0682 .0656 .0631
7 .1145 .1118 .1091 .1064 .1037 .1010 .0982 .0955 .0928 .0901
8 .1302 .1286 .1269 .1251 .1232 .1212 .1191 .1170 .1148 .1126
9 .1317 .1315 .1311 .1306 .1300 .1293 .1284 .1274 .1263 .1251
10 .1198 .1210 .1219 .1228 .1235 .1241 .1245 .1249 .1250 .1251
11 .0991 .1012 .1031 .1049 .1067 .1083 .1098 .1112 .1125 .1137
12 .0752 .0776 .0799 .0822 .0844 .0866 .0888 .0908 .0928 .0948
13 .0526 .0549 .0572 .0594 .0617 .0640 .0662 .0685 .0707 .0729
14 .0342 .0361 .0380 .0399 .0419 .0439 .0459 .0479 .0500 .0521
15 .0208 .0221 .0235 .0250 .0265 .0281 .0297 .0313 .0330 .0347
16 .0118 .0127 .0137 .0147 .0157 .0168 .0180 .0192 .0204 .0217
17 .0063 .0069 .0075 .0081 .0088 .0095 .0103 .0111 .0119 .0128
18 .0032 .0035 .0039 .0042 .0046 .0051 .0055 .0060 .0065 .0071
19 .0015 .0017 .0019 .0021 .0023 .0026 .0028 .0031 .0034 .0037
20 .0007 .0008 .0009 .0010 .0011 .0012 .0014 .0015 .0017 .0019
21 .0003 .0003 .0004 .0004 .0005 .0006 .0006 .0007 .0008 .0009
22 .0001 .0001 .0002 .0002 .0002 .0002 .0003 .0003 .0004 .0004
23 .0000 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0002 .0002
24 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0001 .0001
TABLE 5.8
SELECTED VALUES FROM THE POISSON PROBABILITYTABLES
EXAMPLE: μ ⫽ 10, x ⫽ 5; f(5) ⫽ .0378
The random variable here is x ⫽ number of cars arriving in any 15-minute period.
If management wanted to know the probability of exactly five arrivals in 15 minutes,
we would set x ⫽ 5 and thus obtain
Although this probability was determined by evaluating the probability function with μ ⫽ 10
and x ⫽ 5, it is often easier to refer to a table for the Poisson distribution. The table provides
probabilities for specific values of x and μ. We included such a table as Table 7 of Appen-
dix B. For convenience, we reproduced a portion of this table as Table 5.8. Note that to use
the table of Poisson probabilities, we need know only the values of x and μ. From Table 5.8
we see that the probability of five arrivals in a 15-minute period is found by locating the value
in the row of the table corresponding to x ⫽ 5 and the column of the table corresponding to
μ ⫽ 10. Hence, we obtain f(5) ⫽ .0378.
In the preceding example, the mean of the Poisson distribution is μ ⫽ 10 arrivals per
15-minute period. Aproperty of the Poisson distribution is that the mean of the distribution
and the variance of the distribution are equal. Thus, the variance for the number of arrivals
during 15-minute periods is σ
2
⫽ 10. The standard deviation is σ ⫽
Our illustration involves a 15-minute period, but other time periods can be used.
Suppose we want to compute the probability of one arrival in a 3-minute period. Because
兹
10 ⫽ 3.16.
Probability of exactly
5 arrivals in 15 minutes
⫽ f(5) ⫽
10
5
e
⫺10
5!
⫽ .0378
A property of the Poisson
distribution is that the
mean and variance are
equal.
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220 Chapter 5 Discrete Probability Distributions
10 is the expected number of arrivals in a 15-minute period, we see that 10/15 ⫽ 2/3
is the expected number of arrivals in a 1-minute period and that (2/3)(3 minutes) ⫽ 2
is the expected number of arrivals in a 3-minute period. Thus, the probability of x
arrivals in a 3-minute time period with μ ⫽ 2 is given by the following Poisson
probability function.
The probability of one arrival in a 3-minute period is calculated as follows:
Earlier we computed the probability of five arrivals in a 15-minute period; it was .0378.
Note that the probability of one arrival in a 3-minute period (.2707) is not the same. When
computing a Poisson probability for a different time interval, we must first convert the mean
arrival rate to the time period of interest and then compute the probability.
An Example Involving Length or Distance Intervals
Let us illustrate an application not involving time intervals in which the Poisson
distribution is useful. Suppose we are concerned with the occurrence of major defects
in a highway one month after resurfacing. We will assume that the probability of a
defect is the same for any two highway intervals of equal length and that the occurrence
or nonoccurrence of a defect in any one interval is independent of the occurrence or
nonoccurrence of a defect in any other interval. Hence, the Poisson distribution can be
applied.
Suppose we learn that major defects one month after resurfacing occur at the average
rate of two per mile. Let us find the probability of no major defects in a particular 3-mile
section of the highway. Because we are interested in an interval with a length of 3 miles,
μ ⫽ (2 defects/mile)(3 miles) ⫽ 6 represents the expected number of major defects over the
3-mile section of highway. Using equation (5.11), the probability of no major defects is
f(0) ⫽ 6
0
e
⫺6
/0! ⫽ .0025. Thus, it is unlikely that no major defects will occur in the 3-mile
section. In fact, this example indicates a 1 ⫺ .0025 ⫽ .9975 probability of at least one major
defect in the 3-mile highway section.
Exercises
Methods
38. Consider a Poisson distribution with μ ⫽ 3.
a. Write the appropriate Poisson probability function.
b. Compute f(2).
c. Compute f(1).
d. Compute P(x ⱖ 2).
39. Consider a Poisson distribution with a mean of two occurrences per time period.
a. Write the appropriate Poisson probability function.
b. What is the expected number of occurrences in three time periods?
c. Write the appropriate Poisson probability function to determine the probability of x
occurrences in three time periods.
d. Compute the probability of two occurrences in one time period.
e. Compute the probability of six occurrences in three time periods.
f. Compute the probability of five occurrences in two time periods.
Probability of exactly
1 arrival in 3 minutes
⫽ f(1) ⫽
2
1
e
⫺2
1!
⫽ .2707
f(x) ⫽
2
x
e
⫺2
x!
test
SELF
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Applications
40. Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5-minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes.
c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the
current call, how many callers do you expect to be waiting by that time? What is the
probability that none will be waiting?
d. If no calls are currently being processed, what is the probability that the agent can take
3 minutes for personal time without being interrupted by a call?
41. During the period of time that a local university takes phone-in registrations, calls come in
at the rate of one every two minutes.
a. What is the expected number of calls in one hour?
b. What is the probability of three calls in five minutes?
c. What is the probability of no calls in a five-minute period?
42. More than 50 million guests stay at bed and breakfasts (
B&Bs) each year. The website for
the Bed and Breakfast Inns of North America, which averages seven visitors per minute,
enables many
B&Bs to attract guests (Time, September 2001).
a. Compute the probability of no website visitors in a one-minute period.
b. Compute the probability of two or more website visitors in a one-minute period.
c. Compute the probability of one or more website visitors in a 30-second period.
d. Compute the probability of five or more website visitors in a one-minute period.
43. Airline passengers arrive randomly and independently at the passenger-screening facility
at a major international airport. The mean arrival rate is 10 passengers per minute.
a. Compute the probability of no arrivals in a one-minute period.
b. Compute the probability that three or fewer passengers arrive in a one-minute period.
c. Compute the probability of no arrivals in a 15-second period.
d. Compute the probability of at least one arrival in a 15-second period.
44. An average of 15 aircraft accidents occur each year (The World Almanac and Book of
Facts, 2004).
a. Compute the mean number of aircraft accidents per month.
b. Compute the probability of no accidents during a month.
c. Compute the probability of exactly one accident during a month.
d. Compute the probability of more than one accident during a month.
45. The National Safety Council (NSC) estimates that off-the-job accidents cost U.S. busi-
nesses almost $200 billion annually in lost productivity (National Safety Council, March
2006). Based on NSC estimates, companies with 50 employees are expected to average
three employee off-the-job accidents per year. Answer the following questions for com-
panies with 50 employees.
a. What is the probability of no off-the-job accidents during a one-year period?
b. What is the probability of at least two off-the-job accidents during a one-year
period?
c. What is the expected number of off-the-job accidents during six months?
d. What is the probability of no off-the-job accidents during the next six months?
5.6 Hypergeometric Probability Distribution
The hypergeometric probability distributionis closely related to the binomial distribution.
The two probability distributions differ in two key ways. With the hypergeometric distribu-
tion, the trials are not independent; and the probability of success changes from trial to trial.
test
SELF
5.6 Hypergeometric Probability Distribution 221
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Note that represents the number of ways n elements can be selected from a
population of size N; represents the number of ways that x successes can be selected
from a total of r successes in the population; and represents the number of ways
that n ⫺ x failures can be selected from a total of N ⫺ r failures in the population.
For the hypergeometric probability distribution, x is a discrete random variable and the
probability function f(x) given by equation (5.12) is usually applicable for values of x ⫽ 0,
1, 2,...,n. However, only values of x where the number of observed successes is less than
or equal to the number of successes in the population (x ⱕ r) and where the number of
observed failures is less than or equal to the number of failures in the population
(n ⫺ x ⱕ N ⫺ r) are valid. If these two conditions do not hold for one or more values of x,
the corresponding value of f(x) ⫽ 0, indicating that the probability of this value of x is zero.
To illustrate the computations involved in using equation (5.12), let us consider the
following quality control application. Electric fuses produced by Ontario Electric are pack-
aged in boxes of 12 units each. Suppose an inspector randomly selects 3 of the 12 fuses
in a box for testing. If the box contains exactly 5 defective fuses, what is the probability
that the inspector will find exactly 1 of the 3 fuses defective? In this application, n ⫽ 3
and N ⫽ 12. With r ⫽ 5 defective fuses in the box the probability of finding x ⫽ 1 defec-
tive fuse is
f(1) ⫽
冢
5
1
冣冢
7
2
冣
冢
12
3
冣
⫽
冢
5!
1!4!
冣冢
7!
2!5!
冣
冢
12!
3!9!
冣
⫽
(5)(21)
220
⫽ .4773
冢
N ⫺ r
n ⫺ x
冣
冢
r
x
冣
冢
N
n
冣
222 Chapter 5 Discrete Probability Distributions
HYPERGEOMETRIC PROBABILITY FUNCTION
(5.12)
where
x ⫽
n ⫽
f(x) ⫽
N ⫽
r ⫽
the number of successes
the number of trials
the probability of x successes in n trials
the number of elements in the population
the number of elements in the population labeled success
f(x) ⫽
冢
r
x
冣冢
N ⫺ r
n ⫺ x
冣
冢
N
n
冣
In the usual notation for the hypergeometric distribution, r denotes the number of
elements in the population of size N labeled success, and N ⫺ r denotes the number of
elements in the population labeled failure. The hypergeometric probability function
is used to compute the probability that in a random selection of n elements, selected
without replacement, we obtain x elements labeled success and n ⫺ x elements labeled
failure. For this outcome to occur, we must obtain x successes from the r successes in
the population and n ⫺ x failures from the N ⫺ r failures. The following hypergeo-
metric probability function provides f(x), the probability of obtaining x successes in
n trials.
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5.6 Hypergeometric Probability Distribution 223
Exercises
Methods
46. Suppose N ⫽ 10 and r ⫽ 3. Compute the hypergeometric probabilities for the following
values of n and x.
a. n ⫽ 4, x ⫽ 1
b. n ⫽ 2, x ⫽ 2
c. n ⫽ 2, x ⫽ 0
d. n ⫽ 4, x ⫽ 2
e. n ⫽ 4, x ⫽ 4
47. Suppose N ⫽ 15 and r ⫽ 4. What is the probability of x ⫽ 3 for n ⫽ 10?
Now suppose that we wanted to know the probability of finding at least 1 defective
fuse. The easiest way to answer this question is to first compute the probability that the
inspector does not find any defective fuses. The probability of x ⫽ 0 is
With a probability of zero defective fuses f(0) ⫽ .1591, we conclude that the probability of
finding at least 1 defective fuse must be 1 ⫺ .1591 ⫽ .8409. Thus, there is a reasonably high
probability that the inspector will find at least 1 defective fuse.
The mean and variance of a hypergeometric distribution are as follows.
(5.13)
(5.14)
In the preceding example n ⫽ 3, r ⫽ 5, and N ⫽ 12. Thus, the mean and variance for the
number of defective fuses are
The standard deviation is σ ⫽⫽.77.
兹
.60
σ
2
⫽ n
冢
r
N
冣冢
1 ⫺
r
N
冣冢
N ⫺ n
N ⫺ 1
冣
⫽ 3
冢
5
12
冣冢
1 ⫺
5
12
冣冢
12 ⫺ 3
12 ⫺ 1
冣
⫽ .60
μ ⫽ n
冢
r
N
冣
⫽ 3
冢
5
12
冣
⫽ 1.25
Var(x) ⫽ σ
2
⫽ n
冢
r
N
冣冢
1 ⫺
r
N
冣冢
N ⫺ n
N ⫺ 1
冣
E(x) ⫽ μ ⫽ n
冢
r
N
冣
f(0) ⫽
冢
5
0
冣冢
7
3
冣
冢
12
3
冣
⫽
冢
5!
0!5!
冣冢
7!
3!4!
冣
冢
12!
3!9!
冣
⫽
(1)(35)
220
⫽ .1591
test
SELF
NOTES AND COMMENTS
Consider a hypergeometric distribution with n trials.
Let p ⫽ (r/N) denote the probability of a success on
the first trial. If the population size is large, the term
(N ⫺ n)/(N ⫺ 1) in equation (5.14) approaches 1. As
a result, the expected value and variance can be writ-
ten E(x) ⫽ np and Var(x) ⫽ np(1 ⫺ p). Note that these
expressions are the same as the expressions used to
compute the expected value and variance of a binomial
distribution, as in equations (5.9) and (5.10). When the
population size is large, a hypergeometric distribu-
tion can be approximated by a binomial distribution
with ntrials and a probability of success p ⫽ (r/N).
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224 Chapter 5 Discrete Probability Distributions
Applications
48. In a survey conducted by the Gallup Organization, respondents were asked, “What is your
favorite sport to watch?” Football and basketball ranked number one and two in terms of
preference (Gallup website, January 3, 2004). Assume that in a group of 10 individuals, 7
prefer football and 3 prefer basketball. Arandom sample of 3 of these individuals is selected.
a. What is the probability that exactly 2 prefer football?
b. What is the probability that the majority (either 2 or 3) prefer football?
49. Blackjack, or twenty-one as it is frequently called, is a popular gambling game played in
Las Vegas casinos. A player is dealt two cards. Face cards (jacks, queens, and kings) and
tens have a point value of 10. Aces have a point value of 1 or 11. A 52-card deck contains
16 cards with a point value of 10 (jacks, queens, kings, and tens) and four aces.
a. What is the probability that both cards dealt are aces or 10-point cards?
b. What is the probability that both of the cards are aces?
c. What is the probability that both of the cards have a point value of 10?
d. A blackjack is a 10-point card and an ace for a value of 21. Use your answers to parts
(a), (b), and (c) to determine the probability that a player is dealt blackjack. (Hint:
Part (d) is not a hypergeometric problem. Develop your own logical relationship as to
how the hypergeometric probabilities from parts (a), (b), and (c) can be combined to
answer this question.)
50. Axline Computers manufactures personal computers at two plants, one in Texas and the
other in Hawaii. The Texas plant has 40 employees; the Hawaii plant has 20. A random
sample of 10 employees is to be asked to fill out a benefits questionnaire.
a. What is the probability that none of the employees in the sample work at the plant in
Hawaii?
b. What is the probability that one of the employees in the sample works at the plant in
Hawaii?
c. What is the probability that two or more of the employees in the sample work at the
plant in Hawaii?
d. What is the probability that nine of the employees in the sample work at the plant in
Texas?
51. The Zagat Restaurant Survey provides food, decor, and service ratings for some of the top
restaurants across the United States. For 15 restaurants located in Boston, the average price
of a dinner, including one drink and tip, was $48.60. You are leaving for a business trip to
Boston and will eat dinner at three of these restaurants. Your company will reimburse you
for a maximum of $50 per dinner. Business associates familiar with these restaurants have
told you that the meal cost at one-third of these restaurants will exceed $50. Suppose that
you randomly select three of these restaurants for dinner.
a. What is the probability that none of the meals will exceed the cost covered by your
company?
b. What is the probability that one of the meals will exceed the cost covered by your com-
pany?
c. What is the probability that two of the meals will exceed the cost covered by your com-
pany?
d. What is the probability that all three of the meals will exceed the cost covered by your
company?
52. The Troubled Asset Relief Program (TARP), passed by the U.S. Congress in October
2008, provided $700 billion in assistance for the struggling U.S. economy. Over $200 bil-
lion was given to troubled financial institutions with the hope that there would be an in-
crease in lending to help jump-start the economy. But three months later, a Federal
Reserve survey found that two-thirds of the banks that had received TARP funds had
tightened terms for business loans (The Wall Street Journal, February 3, 2009). Of the 10
banks that were the biggest recipients of TARP funds, only 3 had actually increased lend-
ing during this period.
test
SELF
CH005.qxd 8/16/10 6:31 PM Page 224
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.