Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
Which of the following describes the possible values of a Poisson random variable, X?
1)
A)
All counting numbers (1, 2, 3, 4, …)
B)
All counting numbers up to twice the mean of X (1, 2, 3, 4, …, 2)
C)
All nonnegative integers
D)
All integers
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
2)
Identify each of the variables in the Binomial Probability Formula.
P(x) =n!
(n – x)!x! ·px·(1–p)n–x
Also, explain what the fraction n!
(n – x)!x! computes.
2)
3)
A person is trying to decide which of two possible mutual funds to invest his money in. Let
the random variable X represent the annual return for mutual fund A and let the random
variable Y represent the annual return for fund B. It is known that the mean, µ, of X is
10.3% and the standard deviation, , of X is 4.2%. It is also known that the mean, µ, of Y is
11.3% and the standard deviation, , of Y is 7.2%. Which fund do you think the person
would prefer if he is a short–term investor? Which fund do you think he would prefer if he
is a long–term investor? Explain your thinking.
3)
4)
Suppose that the random variable X has a binomial distribution and that the success
probability, p, is greater than 0.5. Is the probability distribution of X right skewed, left
skewed, or symmetric? Explain your thinking.
4)
5)
Suppose a mathematician computed the expected value of winnings for a person playing
each of seven different games in a casino. What would you expect to be true for all
expected values for these seven games?
5)
6)
The random variable X represents the number of thunderstorms occurring in a month in
one city. Suppose that X has a Poisson distribution with parameter = 3.2. Determine and
interpret the mean of the random variable X.
6)
7)
Let the random variable X represent the winnings at one play of a particular game. The
expected value of X is known to be –$0.32. Suppose a player plays the game five times and
calculates his average winnings. Will the average definitely be equal to –$0.32? Now
suppose the player plays the game 100 times and calculates his average winnings. Will the
average definitely be equal to –$0.32? Which average is likely to be closer to –$0.32?
Explain your answer with reference to the law of large numbers.
7)
8)
A group of potential jurors consists of 15 women and 18 men. Suppose that 12 people are
picked at random from this group, without replacement. Let X represent the number of
women among those selected. Since the sample size exceeds 5% of the population size, X
does not have an approximate binomial distribution. Explain in your own words why X
does not have a binomial distribution. Which of the requirements for a binomial
distribution does it not satisfy?
8)
9)
Give an example of a discrete random variable whose possible values form a countable
infinite set of numbers.
9)
Solve the problem.
10)
11% of the employees of a certain company cycle to work. Three employees are selected at
random from the company and asked whether or not they cycle to work. Considering a
success to be “cycles to work”, formulate the process of observing whether each of the three
employees cycles to work as a sequence of three Bernoulli trials. Complete the table below
by showing each possible outcome together with its probability. Display the probabilities
to three decimal places. List the outcomes in which exactly one of the three employees
cycles to work. Without using the binomial probability formula, find the probability that
exactly one of the three employees cycles to work.
Outcome Probability
sss (0.11)(0.11)(0.11) =0.001
10)
11)
The number of lightning strikes in a year at the top of a particular mountain has a Poisson
distribution
with parameter = 3.5. Construct a histogram of the probabilities when the number of
strikes is from 1–5.
11)
Provide an appropriate response.
12)
Let the random variable X represent the winnings at one play of game A. The mean, µ, of X
is known to be –$0.42 and its standard deviation, , is $0.27. Let the random variable Y
represent the winnings at one play of game B. The mean, µ, of Y is known to be –$0.42 and
its standard deviation, , is $0.20. You have decided to play one of these two games just
once. At which game are you more likely to make a profit (i.e., to not lose money)? Explain
your thinking.
12)
13)
A coin is biased. Danny wishes to determine the probability of obtaining heads when
flipping this coin. He flips the coin 10 times and obtains 8 heads. He concludes that the
probability of obtaining heads when flipping this coin is 0.8. Is his thinking reasonable?
Why or why not?
13)
Solve the problem.
14)
30% of the adult residents of a certain city own their own home. Four residents are
selected at random from the city and asked whether or not they own their own home.
Considering a success to be “owns their own home”, formulate the process of observing
whether each of the four residents owns their own home as a sequence of four Bernoulli
trials. Complete the table below by showing each possible outcome together with its
probability. Display the probabilities to three decimal places. List the outcomes in which
exactly two of the four residents own their own home. Without using the binomial
probability formula, find the probability that exactly two of the four residents own their
own home.
Outcome Probability
ssss (0.3)(0.3)(0.3)(0.3) =0.008
14)
Provide an appropriate response.
15)
Explain how you would construct a probability histogram of a discrete random variable
given its probability distribution.
15)
Solve the problem.
16)
A coin is biased so that the probability it will come up tails is 0.44. The coin is tossed three
times. Considering a success to be tails, formulate the process of observing the outcome of
the three tosses as a sequence of three Bernoulli trials. Complete the table below by
showing each possible outcome together with its probability. Display the probabilities to
three decimal places. List the outcomes in which exactly two of the three tosses are tails.
Without using the binomial probability formula, find the probability that exactly two of the
three tosses are tails.
Outcome Probability
hhh (0.56)(0.56)(0.56) =0.176
16)
Provide an appropriate response.
17)
List the four requirements for a binomial distribution. Describe an experiment which is
binomial and discuss how the experiment fits each of the four requirements.
17)
18)
6.2% of VCRs of a certain type are defective. Let the random variable X represent the
number of defective VCRs among 200 randomly selected VCRs of this type. Suppose you
wish to find the probability that X is equal to 8. Does the random variable X have a
binomial or a Poisson distribution? How can you tell? If X has a binomial distribution,
would it be reasonable to use the Poisson approximation? If not, why not?
18)
19)
A game is said to be “fair” if the expected value for winnings is 0, that is, in the long run,
the player can expect to win 0. Consider the following game. The game costs $1 to play and
the payoffs are $5 for red, $3 for blue, $2 for yellow, and nothing for white. The following
probabilities apply. What are your expected winnings? Does the game favor the player or
the owner?
Outcome Probability
Red .02
Blue .04
Yellow .16
White .78
19)
20)
Describe the Poisson distribution and give some example of a random variable with a
Poisson distribution.
20)
21)
For a particular game at a casino, let the random variable X represent the winnings (payoff
minus bet) for one play of the game. The expected value of the random variable X is
–$0.87. How would you interpret this statement?
21)
22)
Five cards are drawn at random, with replacement, from an ordinary deck of 52 cards.
Considering success to be drawing a heart, formulate the process of observing the suits of
the five cards as a sequence of five Bernoulli trials.
22)
23)
Explain in your own words the meaning of the term “probability distribution”.
23)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
24)
Let the random variable X represent the winnings at one play of a particular game. The expected
value of X is known to be –$0.32. True or false, this means that on any given play of the game, the
most likely outcome is that the player will lose 32 cents?
24)
A)
True
B)
False
Find the indicated probability. Round to four decimal places.
25)
In a study, 36% of adults questioned reported that their health was excellent. A researcher wishes to
study the health of people living close to a nuclear power plant. Among 10 adults randomly
selected from this area, only 3 reported that their health was excellent. Find the probability that
when 10 adults are randomly selected, 3 or fewer are in excellent health.
25)
A)
0.4868
B)
0.2462
C)
0.2405
D)
0.3521
26)
A test consists of 10 true/false questions. To pass the test a student must answer at least 7 questions
correctly. If a student guesses on each question, what is the probability that the student will pass
the test?
26)
A)
0.1719
B)
0.0547
C)
0.1172
D)
0.9453
27)
An airline estimates that 92% of people booked on their flights actually show up. If the airline
books 77 people on a flight for which the maximum number is 75, what is the probability that the
number of people who show up will exceed the capacity of the plane?
27)
A)
0.0109
B)
0.0125
C)
0.0485
D)
0.0016
Find the mean of the binomial random variable. Round to two decimal places when necessary.
28)
The probability that a person has immunity to a particular disease is 0.8. Find the mean for the
random variable X, the number who have immunity in samples of size 24.
28)
A)
0.8
B)
4.8
C)
19.2
D)
12
Calculate the specified probability
29)
Suppose that T is a random variable. Given that P(2.05 T 2.05) =0.575, and that P(K <2.05) =
P(K >2.05), find P(K < – 2.05).
29)
A)
0.425
B)
1.025
C)
0.575
D)
0.2125
Find the indicated probability. Round to four decimal places.
30)
A company purchases shipments of machine components and uses this acceptance sampling plan:
Randomly select and test 26 components and accept the whole batch if there are fewer than 3
defectives. If a particular shipment of thousands of components actually has a 7% rate of defects,
what is the probability that this whole shipment will be accepted?
30)
A)
0.2790
B)
0.7272
C)
0.5756
D)
0.1680
Calculate the specified probability
31)
Suppose that K is a random variable. Given that P(–2.2
K 2.2) =0.725, and that P(K < – 2.2) = P(K
>2.2), find P(K >2.2).
31)
A)
0.1375
B)
0.275
C)
1.1
D)
0.725
A
Evaluate the expression.
32)
9!
32)
A)
40,320
B)
362,880
C)
362,889
D)
362,871
B
Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials.
Let X denote the total number of successes. Round to three decimal places.
33)
n =5, p =1
4, P(X =3)
33)
A)
0.114
B)
0.016
C)
0.088
D)
0.132
C
Provide an appropriate response.
34)
True or false? For any discrete random variable, the possible values of the random variable form a
finite set of numbers.
34)
A)
True
B)
False
B
B
Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
35)
The number of power failures experienced by the Columbia Power Company in a day has a
Poisson distribution with parameter = 0.210. Find the probability that there are exactly two
power failures in a particular day.
35)
A)
0.085
B)
0.018
C)
0.027
D)
0.036
Find the mean of the binomial random variable. Round to two decimal places when necessary.
36)
According to a college survey, 22% of all students work full time. Find the mean for the random
variable X, the number of students who work full time in samples of size 16.
36)
A)
3.52
B)
0.22
C)
4
D)
2.75
A
Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
37)
In a game, you have a 1/37 probability of winning $69 and a 36/37 probability of losing $2. What is
your expected value?
37)
A)
–$1.95
B)
–$0.08
C)
$3.81
D)
$1.86
B
Find the standard deviation of the Poisson random variable. Round to three decimal places.
38)
The number of calls received by a car towing service in an hour has a Poisson distribution with
parameter =2.100. Let X denote the number of calls received by the service in a randomly selected
hour. Find the standard deviation of X.
38)
A)
1.050
B)
4.410
C)
1.449
D)
2.100
C
Provide an appropriate response.
39)
True or false, a Poisson random variable has an infinite number of possible values?
39)
A)
True
B)
False
A
B
Obtain the probability distribution of the random variable.
40)
The following table displays a frequency distribution for the number of siblings for students at one
middle school. For a randomly selected student in the school, let X denote the number of siblings of
the student. Obtain the probability distribution of X.
Number of siblings 0 1 2 3 4 5 6 7
Frequency 182 243 124 53 25 9 7 1
40)
A)
Siblings
x
Probability
P(X = x)
00.283
10.377
20.193
30.082
40.039
50.014
60.011
70.002
B)
Siblings
x
Probability
P(X = x)
00.125
10.125
20.125
30.125
40.125
50.125
60.125
70.125
C)
Siblings
x
Probability
P(X = x)
10.526
20.268
30.115
40.054
50.019
60.015
70.002
D)
Siblings
x
Probability
P(X = x)
00.298
10.362
20.205
30.070
40.039
50.014
60.011
70.002
Determine the possible values of the random variable.
41)
The following table displays a frequency distribution for the number of siblings for students in one
middle school. For a randomly selected student in the school, let X denote the number of siblings of
the student. What are the possible values of the random variable X?
Number of siblings 0 1 2 3 4 5 6 7
Frequency 189 245 102 42 24 13 5 2
41)
A)
7
B)
189, 245, 102, 42, 24, 13, 5, 2
C)
0, 1, 2, 3, 4, 5, 6, 7
D)
Brother, sister
C
A
Find the mean of the binomial random variable. Round to two decimal places when necessary.
42)
In a certain town, 60 percent of voters are in favor of a given ballot measure and 40 percent are
opposed. For groups of 140 voters, find the mean for the random variable X, the number who
oppose the measure.
42)
A)
84
B)
40
C)
60
D)
56
Find the mean of the random variable.
43)
The random variable X is the number of golf balls ordered by customers at a pro shop. Its
probability distribution is given in the table. Round the answer to two decimal places when
necessary.
x 3 6 9 12 15
P(X = x) 0.14 0.21 0.36 0.19 0.10
43)
A)
8.82
B)
9
C)
8.7
D)
6.03
C
Find the mean of the Poisson random variable.
44)
The number of calls received by a car towing service in an hour has a Poisson distribution with
parameter =2.49. Let X denote the number of calls received by the service in a randomly selected
hour. Find the mean of X.
44)
A)
2.49
B)
1.245
C)
1.578
D)
6.2
A
Evaluate the expression.
45)
(16 –7)!
45)
A)
40,320
B)
3,628,800
C)
362,880
D)
9
C
D
Find the mean of the binomial random variable. Round to two decimal places when necessary.
46)
The probability is 0.4 that a person shopping at a certain store will spend less than $20. For groups
of size 23, find the mean number who spend less than $20.
46)
A)
8
B)
12
C)
13.8
D)
9.2
Use random–variable notation to represent the event.
47)
Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the
two numbers. Use random–variable notation to represent the event that the absolute value of the
difference of the two numbers is 2.
47)
A)
P{X = 2}
B)
X = 2
C)
{(1, 3), (2, 4), (3, 5), (4, 6), (3, 1), (4, 2), (5, 3), (6, 4)}
D)
{X = 2}
D
Calculate the specified probability
48)
Suppose that W is a random variable. Given that P(W 3) =0.625, find P(W >3).
48)
A)
0.625
B)
3
C)
0
D)
0.375
D
D
Construct a probability histogram for the binomial random variable, X.
49)
A baseball player batting 0.300 comes to bat 4 times in a game. X is the number of hits.
49)
A)
B)
C)
D)
Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
50)
Suppose you pay $3.00 to roll a fair die with the understanding that you will get back $5.00 for
rolling a 6 or a 4, nothing otherwise. What is your expected value?
50)
A)
$3.00
B)
–$3.00
C)
–$1.33
D)
$5.00
Find the indicated binomial probability. Round to five decimal places when necessary.
51)
A multiple choice test has 30 questions, and each has four possible answers, of which one is correct.
If a student guesses on every question, find the probability of getting exactly 18 correct.
51)
A)
0.00126
B)
0.02906
C)
0.00004
D)
255,257,450,754
Provide an appropriate response.
52)
The random variable X represents the number of siblings of a student selected randomly from a
particular college. Use random variable notation to express the following statement in shorthand.
The probability that the student has two siblings is 0.18.
52)
A)
(X = 2) = 0.18
B)
P(X = 2) = 0.18
C)
P(X) = 0.18
D)
P(2) = 0.18
Find the mean of the Poisson random variable.
53)
Suppose X has a Poisson distribution with parameter =0.63. Find the mean of X.
53)
A)
0.794
B)
0.63
C)
0.4
D)
0.315
Solve the problem.
54)
A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a
Poisson distribution with parameter = 3.3. Construct a probability table for the random variable
X. Compute the probabilities for 0 – 5 sightings.
54)
A)
xP(X=x)
0 0.370
1 0.122
2 0.201
3 0.221
4 0.182
5 0.120
B)
xP(X=x)
0 0.037
1 0.122
2 0.201
3 0.221
4 0.182
5 0.120
C)
xP(X=x)
0 0.037
1 0.122
2 0.201
3 0.221
4 0.182
5 0.012
D)
xP(X=x)
0 0.037
1 0.122
2 0.021
3 0.221
4 0.182
5 0.120
Find the indicated binomial probability. Round to five decimal places when necessary.
55)
A company manufactures calculators in batches of 64 and there is a 4% rate of defects. Find the
probability of getting exactly 5 defects in a batch.
55)
A)
685,827.352
B)
0.07023
C)
4.8
D)
0.78075
Find the mean of the binomial random variable. Round to two decimal places when necessary.
56)
A company manufactures batteries in batches of 7 and there is a 3% rate of defects. Find the mean
for the random variable X, the number of defects per batch.
56)
A)
6.79
B)
0.2
C)
0.22
D)
0.21
Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
57)
Sue Anne owns a medium–sized business. Use the probability distribution below, where X
describes the number of employees who call in sick on a given day.
Number of Employees Sick 0 1 2 3 4
P(X = x) 0.1 0.35 0.25 0.25 0.05
What is the expected value of the number of employees calling in sick on any given day? Round
the answer to two decimal places.
57)
A)
1.90
B)
1.80
C)
1.00
D)
2.00
The probability distribution of a random variable is given along with its mean and standard deviation. Draw a
probability histogram for the random variable; locate the mean and show one, two, and three standard deviation
intervals.
58)
x 4 5 6 7 8
P(X = x) 0.1 0.3 0.45 0.1 0.05
µ= 5.7, = 0.95
58)
A)
16
B)
C)
Determine the required probability by using the Poisson approximation to the binomial distribution. Round to three
decimal places.
59)
The probability that a car will have a flat tire while driving through a certain tunnel is 0.00006. Use
the Poisson approximation to the binomial distribution to find the probability that among 13,000
cars passing through this tunnel, at least one will have a flat tire.
59)
A)
0.542
B)
0.358
C)
0.458
D)
0.965
Find the standard deviation of the random variable.
60)
The random variable X is the number of houses sold by a realtor in a single month at the Sendsom’s
Real Estate office. Its probability distribution is given in the table. Round the answer to two decimal
places.
Houses Sold (x) 0 1 2 3 4 5 6 7
Probability P(x) 0.24 0.01 0.12 0.16 0.01 0.14 0.11 0.21
60)
A)
6.86
B)
2.25
C)
4.45
D)
2.62
Find the specified probability.
61)
A statistics professor has office hours from 9:00 am to 10:00 am each day. The number of students
waiting to see the professor is a random variable, X, with the distribution shown in the table.
x 0 1 2 3 4 5
P(X = x) 0.05 0.10 0.40 0.25 0.15 0.05
The professor gives each student 10 minutes. Determine the probability that a student arriving just
after 9:00 am will have to wait at least 20 minutes to see the professor.
61)
A)
0.45
B)
0.25
C)
0.85
D)
0.40
Find the specified probability distribution of the binomial random variable.
62)
In one city, the probability that a person will pass his or her driving test on the first attempt is 0.65.
Four people are selected at random from among those taking their driving test for the first time.
Determine the probability distribution of X, the number among the four who pass the test.
62)
A)
x P(X = x)
10.65
20.4225
30.2746
40.1785
B)
x P(X = x)
00.0150
10.1115
20.3105
30.3844
40.1785
C)
x P(X = x)
00.0150
10.1115
20.3105
30.4024
40.1605
D)
x P(X = x)
00.0150
10.1115
20.2070
30.3844
40.1785
Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
63)
The number of calls received by a mountain search and rescue team in a day has a Poisson
distribution with parameter =0.52. Find the probability that on a randomly selected day, they
will receive fewer than two calls.
63)
A)
0.904
B)
0.08
C)
0.309
D)
0.096
Find the indicated binomial probability. Round to five decimal places when necessary.
64)
In a certain college, 20% of the physics majors belong to ethnic minorities. If 10 students are
selected at random from the physics majors, what is the probability that exactly 3 belong to an
ethnic minority?
64)
A)
0.20133
B)
0.96
C)
0.00079
D)
0.00168
Determine the binomial probability formula given the number of trials and the success probability for Bernoulli trials.
Let X denote the total number of successes. Round to three decimal places.
65)
n =5, p =0.3, P(X =3)
65)
A)
0.172
B)
0.132
C)
0.027
D)
0.198
Use random–variable notation to represent the event.
66)
The following table displays a frequency distribution for the number of siblings for students in one
middle school. For a randomly selected student in the school, let Y denote the number of siblings of
the student.
Number of siblings 0 1 2 3 4 5 6 7
Frequency 189 245 102 42 24 13 5 2
Use random–variable notation to represent the event that the student obtained has at least two but
fewer than six siblings.
66)
A)
{2 Y < 6}
B)
{2 Y 6}
C)
{2 < Y < 6}
D)
{2, 3, 4, 5}
Find the mean of the binomial random variable. Round to two decimal places when necessary.
67)
A die is rolled 10 times and the number of times that two shows on the upper face is counted. If this
experiment is repeated many times, find the mean for the random variable X, the number of twos.
67)
A)
8.33
B)
2.5
C)
3.33
D)
1.67
D)
Determine the possible values of the random variable.
68)
The following frequency distribution lists the annual household incomes (in thousands of dollars)
of one neighborhood in a large city. For a randomly selected income between $200,000 and
$700,000, let Y denote the number of households with that income. What are the possible values of
the random variable Y?
Incomes Frequency
200–300 66
301–400 58
401–500 87
501–600 71
601–700 16
68)
A)
16
B)
66, 58, 87, 71
C)
66, 58, 87, 71 , 16
D)
298
D)
Determine the required probability by using the Poisson approximation to the binomial distribution. Round to three
decimal places.
69)
The rate of defects among CD players of a certain brand is 1.3%. Use the Poisson approximation to
the binomial distribution to find the probability that among 230 such CD players received by a
store, there is at most one defective.
69)
A)
0.201
B)
0.950
C)
0.150
D)
0.799
D)
70)
The probability that a call received by a certain switchboard will be a wrong number is 0.02. Use
the Poisson approximation to the binomial distribution to find the probability that among 140 calls
received by the switchboard, there are at least two wrong numbers.
70)
A)
0.238
B)
0.531
C)
0.769
D)
0.231