Exercises For Chapter 5 (Discrete Prob. Distributions)
Part I Multiple Choice: Select the best answer
For questions: 1-3
500 families were interviewed and the number of children per family was determined. The data
were summarized as follows:
Number of children per family (x)
0
1
2
3
Number of families
55
a
315
50
1. Find the value of "a".
a. 1
b. 185
c. 80
d. 17
e. 420
2. What is the total number of children included in this study? (total children in the 500 families)
a. 80
b. 500
c. 420
d. 860
e. 445
3. If you randomly select a family from the 500 families interviewed, what is the probability that
the selected family has 3 children?
a. 0.03
b. 0.06
c. 0.50
d. 0.80
e. 0.10
4. Suppose 60% of a large group of animals is infected with a particular disease. Let Y= the
number of non- infected animals in a sample of size 5. The distribution of Y is
a. binomial with n = 5 and p = 0.6
b. binomial with n = 5 and p = 0.4
c. binomial with n = 5 and p = 0.5
d. Poisson with   0.6
5. Which of the following best describes the expected value of a discrete random variable?
a. It is the middle value of all possible outcomes
b. It is the weighted average over all possible outcomes.
c. It is the simple average over all possible outcomes
d. None of the above
6. Whenever p = 0.9 and n = 10, the mean of the binomial random variable will be
a. 0.9
b. 10
c. 9
d. 0.95
7. The number of traffic accidents in a small city has a rate (average) of 3 accidents per week .
What is the probability of at least one accident in 2 weeks?
a. 0.0174
b. 0.9502
c. 0.9975
d. 0.1991
8. The random variable X has the following distribution:
X
1
2
P(x)
0.3
0.2
Find P(X = 10)
a. 0.2
b. 0.5
c. 0.3
4
0.2
10
?
d. 0.1
9. For problem (8), the mean and standard deviation of X are, respectively,
a. 1.5, 14.05
b. 4.5, 14.05
c. 4.5, 3.75
d. 0, 3.0
10. For problem (8), find the probability that X exceeds 2.
a. 0
b. 0.7
c. 0.2
1
d. 0.5
11. The number of telephone calls that pass through a switchboard has a Poisson distribution with
mean equal to 2 per minute. The expected number of phone calls that pass through the
switchboard in one minute is
a. 2
b. 4
c. 3
d. 1
12. Refer to question 11. The probability that no telephone calls pass through the switchboard in
two consecutive minutes is
a. 0.2707
b. 0.0517
c. 0.0183
d. 0.0366
13. A class consists of 45 students. Ten of these students received an "A" for the final exam. Five
students are selected at random from this group. Using the Hyper-geometric distribution what is
the probability that at least one of the five students selected received an "A" for the final exam?
a. 0.6993
b. 0.7343
c. 0.2657
d. 0.2675
e. 0.4286
.
14. Consider the following probability distribution:
P x  
5!
0.7x 0.35 x , x  0, 1, ..., 5
x! 5  x !
Find the mean of the random variable x.
a. 5
b. 35
c. 0.21
d. 1.5
e. 3.5
15. It is known that cars arrive at a drive-through at a rate of three cars per minute between 12
noon and 1:00 pm. Assuming the number of cars that arrive in any time interval has a Poisson
distribution, what is the probability that exactly 9 cars arrive between 12:10 and 12:15
a.
e 9 3 9
3!
b.
e 3153
.
15!
c.
e 9 159
9!
d.
e 1515 9
9!
e.
e 3 3 9
9!
16. A new treatment is developed. It is said that the probability of a randomly selected patient
being cured by this treatment is 0.75. If 5 patients receive this treatment, what is the probability
that at least 4 of them will be cured?
a. 0.3955
b. 0.6328
c. 0.2373
d. 0.8000
e. 0.0791
17. The number of traffic accidents per day on a certain section of highway is assumed to be
Poisson with a mean equal to 4. Based on this, how many traffic accidents should be expected
during any give week?
a. 2
b. 4
c. 28
d. 16
e. 20
18. In a certain communication system, there is an average of 1 transmission error per 10 seconds.
Let the distribution of transmission errors be Poisson. What is the probability of having 2 errors
in one-half minute in duration?
a. 0.184
b. 0.224
c. 0.448
d. 0.0498
e. 0.378
19. A company has bought 20 machines from a manufacturer. The manufacturer advises them
that 8 of these machines have a defect. They take a random sample of 5 machines. What is the
probability that exactly 2 of the machines in the sample have a defect?
a. 0.400
b. 0.148
c. 0.625 d. 0.297
e. 0.397
20. For a binomial distribution
a. n must assume a number between 1 and 20.
c. There must be at least 3 possible outcomes
b. P must be multiple of 0.10
d. None of the above
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21. A multiple choice test has 20 questions, with each question having 5 possible answers.
Suppose a student randomly guesses the answer of each question. What is the probability that the
student will answer all 20 questions correctly?
a.
1
5
b.
1
5 20
c.
1
d.
C5
20
1
C5
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22. When sampling without replacement from a finite population, the data follow
a. a normal distribution
b. a binomial distribution
c. a Poisson distribution
d. a hypergeometric distribution
23. The discrete probability distribution that may be used to compute the probability of
occurrence of a random event over some particular time period would be the ________
distribution
a. binomial
b. Poisson
c. hypergeometric
d. none of the above
24. A particular company has twenty salespeople. In how many ways can a group of three
salespeople be selected from this company?
a. 8000
b. 6840
c. 5700
d. 1140
e. 2210
25. Which of the following is not true concerning discrete probability distribution?
a. The probability of any specific value is between 0 and 1, inclusive.
b. The mean of the distribution is between the smallest and largest value of the discrete random
variable.
c. The sum of all probabilities is 1.
d. The standard deviation of the distribution is between -1 and 1.
e. The distribution may be displayed using a probability histogram.
26. A business evaluates a proposed venture as follows. It stands to make a profit of $10,000 with
probability 3/20, to make profit of $5,000 with probability 9/20, to break even with probability
1/4 and to lose $5,000 with probability 3/20. The expected profit in dollars is: i.e.
X (profit)
P(X) probability
a. 1,500
b. 0
Profit
10,000
3/20
c. -1,500
Profit
5,000
9/20
Break even
0
1/4
d. 3,250
Loose
-5,000
3/20
e. 3,000
27. An analyst estimates that a stock has the following probabilities of return depending on the
state of the economy.
Economy
Return
Probability
Good
15%
0.1
Normal
13%
0.6
Poor
7%
0.3
The expected return of the stock is:
a. 7.8%
b. 11.4%
c. 11.7%
d. 13.0%
28. Which of the following statements is true for a continuous random variable X?
a. P(X = c) = 0 for any constant c
b. P(X = c) = 0 only for c = 0
c. the total area under the curve equals one
d. Both (a) and (c)
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29. Which of the following is NOT an assumption of the binomial experiment?
a. All trials must be identical and independent
b. Each trial outcomes must be classified as a success or a failure.
c. The probability of success is equal to 0.5 in all trials
d. The number of successes in the trials is counted.
30. The probability that a certain machine will produce a defective item is 0.20. If a random
sample of 6 items is taken, what is the probability that there will be 5 or more defectives in the
sample?
a. 0.0016
b. 0.0154
c. 0.2458
d. 0.0001
Consider the following situation to answer questions 31-32
According to a survey, 75% of all customers will return to the same grocery store. Suppose eight
customers are selected at random,
31. what is the probability that exactly five of the customers will return?
a. 0.0467
b. 0.2076
c. .2541
d. 0.1468
32. How many customers would be expected to return to the same store?
a. 8
b. 5
c. 6
d. 7
34. The distribution of the number of people logging into a large computer network during a five
second period is
a. binomial
b. Hypergeometric
c. Poisson
d. None of the above
35. A large proportion of small businesses fail during the first few years of operation. On average,
1.3 businesses fail per day in a large city. What is the probability that 3 businesses will fail on a
given day in this city?
a. 0.0998
b. 0.2330
c. 0.1098
d. 0.3110
36. Suppose a professor randomly selects three new teaching assistants from a total of 10
applicants, six male and four female students. The probability that no females are hired is
a. 0.2500
b. 0.333
c. 0.200
d. 0.1667
37. The business department at a university has 18 faculty members. Of them, 11 are in favor of
the proposition that all MBA students should take a course in ethics and 7 are against this
proposition. If 5 faculty members are randomly selected from 18, what is the probability that the
number of faculty members in this sample who are in favor of the proposition is exactly two?
a. 0.2247
b. 0.2696
c. 0.0539
d. 0.3315
38. The number of customers entering a bank per minute is a Poisson random variable with a
mean of 3.5 customers per minute. What is the probability that more than three customers enter
the bank in a minute?
a. 0.3209
b. 0.4633
c. 0.5367
d. 0.6791
39. The marketing manager of a company usually receives 10 complaint calls
during a week (consisting of five working days). Suppose that the number of calls during a week
follows the Poisson distribution. The probability that she gets five such calls in one day is:
a. 0.0361
b. 0.0378
c. 0.9834
d. 0.2000
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Solve the following questions
1. The number of connections on the internet during two-minute period is given by the following
distribution
Number of times
0
1
2
3
4
Proportion
0.1
0.2
0.1
0.4
0.2
a. Determine the mean number of times a connection is made during a two-minute period
b. Determine the standard deviation of the number of times a connection is made during a twominute period
2 Abdullah is considering an investment in a company. He is interested in buying a number of
shares of this company and keep them for one year. The price of a single share at this time is $16
and the forecasted price a year from today is given in the following table
Price of stock in one year
15
16
17
18
19
Probability
0.15
0.20
0.26
0.21
0.18
a. Does the information above describe a valid probability distribution function? Justify
b. What is the variance and standard deviation of the stock price a year from today?
3. A packing company is inspecting a shipment of grapefruit by cutting 15 grapefruit selected at
random. The company will accept all grapefruit if 14 or all of these 15 grapefruit are of
satisfactory quality. Suppose that 95% of the grapefruit of the shipment are of satisfactory
quality.
(a) Write the probability distribution of the grapefruits that are of satisfactory quality in
the sample of 15 grapefruit.
(b) What is the probability that the shipment will be accepted? (i.e. the probability of 14 or 15
good grapefruits out of 15)?
4. A bag contains 7 English books and 5 Math books. Suppose that you randomly select 4 books
from the bag. Let X be the number of Math books in these 4 books that will be selected from the
bag.
a. Write the probability distribution of X?
b. What is the probability that you select two Math and two English books?
5. Ahmad must select two Management courses and 3 Math courses. If the catalog contains 5
management and 8 Math courses, how many ways can Ahmad complete his schedule?
6. Customers use an automatic teller machine at an average of 15 per hour.
a. What is the probability that exactly 12 will use the machine in the next hour?
b. What is the probability that at least one will use the machine in the next 20 minutes?
7. In a statistics class with 15 males and 13 females, five students are selected to put problems on
the board. What is the probability that?
a. 4 females are selected?
b. at most one male is selected?
8. A retailer of electronic equipment received nine cassettes from the manufacturer. Three of the
cassettes were damaged in the shipment. The retailer sold three cassettes to three customers. Let
X denotes the number of damaged cassettes sold to the three customers.
a. Write the probability function of X.
b. What is the probability that two of the three customers received damaged cassettes?
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c. What is the probability that not more than one of the three customers received damaged
cassettes?
9. The following table summarizes investment outcomes (in $1000) and corresponding
probabilities for a particular oil well:
x = the outcomes (in $1000)
P(x)
- 4 (no oil)
0.25
1
0.70
7
0.05
Total
1
a. Find the expected value of the investment outcomes
b. Find the standard deviation for the investment outcomes
10. In a market study, a researcher found that 70% of customers are repeat customers. If 10
customers are selected at random, find the probability that
a. at least one customer is repeat customer.
b. exactly 7 are repeat customers.
c. How many would you expect to be repeat customers?
11. A committee of three people is to be selected from a group of five men and three women.
a. Find the number of possible committees.
b. Find the number of possible committees with two women and one man.
c. Find the probability that at least two men are on the committee.
12. A retail store aimed to reduce the number of bad checks cashed by its cashiers. The store's
goal is to cash no more than eight bad checks per week. The average number of bad checks
cashed is three per week. Let x denote the number of bad checks cashed per week. Assuming that
x has a Poisson distribution:
a. Find the probability that the store's cashiers will not cash any bad checks in a particular week.
b. Find the probability that the store will not meet its goal in a particular week.
c. Find the probability that the store's cashiers will cash no more than 4 bad checks per two-week
period.
d. Find the mean, variance, and standard deviation of the number of bad checks per week?
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