Download Ch. 4 Discrete Random Variables 4.1 Two Types of Random Variables

Ch. 4 Discrete Random Variables
4.1 Two Types of Random Variables
1 Classify Variables as Continuous or Discrete
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Classify the following random variable according to whether it is discrete or continuous.
The number of cups of coffee sold in a cafeteria during lunch
A) discrete
B) continuous
2) Classify the following random variable according to whether it is discrete or continuous.
The height of a player on a basketball team
A) continuous
B) discrete
3) Classify the following random variable according to whether it is discrete or continuous.
The blood pressures of a group of students the day before the final exam
A) continuous
B) discrete
4) Classify the following random variable according to whether it is discrete or continuous.
The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska
A) continuous
B) discrete
5) Classify the following random variable according to whether it is discrete or continuous.
The number of goals scored in a soccer game
A) discrete
B) continuous
6) Classify the following random variable according to whether it is discrete or continuous.
The speed of a car on a Los Angeles freeway during rush hour traffic
A) continuous
B) discrete
7) Classify the following random variable according to whether it is discrete or continuous.
The number of phone calls to the attendance office of a high school on any given school day
A) discrete
B) continuous
8) Classify the following random variable according to whether it is discrete or continuous.
The number of pills in a container of vitamins
A) discrete
B) continuous
9) 50 students were randomly sampled and asked questions about their exercise habits. One of the questions they
were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week.
This variable would be characterized as which type of random variable?
A) discrete
B) continuous
10) The school newspaper surveyed 100 commuter students and asked two questions. First, students were asked
how many courses they were currently enrolled in. Second, the commuter students were asked to estimate how
long it took them to drive to campus. Considering these two variables, number of courses would best be
considered a _________ variable and drive time would be considered a _________ variable.
A) discrete; continuous
B) discrete; discrete
C) continuous; continuous
D) continuous; discrete
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11) Management at a home improvement store randomly selected 95 customers and observed their shopping
habits. They recorded the number of items each of the customers purchased as well as the total time the
customers spent in the store. Identify the types of variables recorded by the managers of the home
improvement store.
A) number of items discrete; total time continuous
B) number of items continuous; total time continuous
C) number of items continuous; total time discrete
D) number of items discrete; total time discrete
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) A coin is flipped 6 times. The variable x represents the number of tails obtained.
List the possible values of x. Is x discrete or continuous? Explain.
13) A bottle contains 16 ounces of water. The variable x represents the volume, in ounces, of water remaining in the
bottle after the first drink is taken. What are the natural bounds for the values of x? Is x discrete or continuous?
Explain.
4.2 Probability Distributions for Discrete Random Variables
1 Construct Probability Distribution
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is
shown below. Find the probability for the value of x 5.
x
2
3
5
8
10
p(x) 0.10 0.20 ??? 0.30 0.10
A) 0.7
B) 0.1
C) 0.3
D) 0.2
2) The Fresh Oven Bakery knows that the number of pies it can sell varies from day to day. The owner believes
that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies
on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at
a cost of $2.50 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies
for $3 each, find the probability distribution for her daily profit.
B)
C)
D)
A)
Profit P(profit)
Profit P(profit)
Profit P(profit)
Profit P(profit)
$200 .5
$300
.5
$100
.5
$50
.5
$50 .25
$450
.25
$250
.25
$75
.25
$100
.25
$600
.25
$400
.25
$100
.25
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3) Explain why the following is or is not a valid probability distribution for the discrete random variable x.
x
p(x)
1
.1
3
.1
5
.2
7
.1
9
.2
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4) Explain why the following is or is not a valid probability distribution for the discrete random variable x.
x
p(x)
1
.1
0
.2
1
.3
2
.3
3
.1
5) Explain why the following is or is not a valid probability distribution for the discrete random variable x.
x
p(x)
0
.1
2
.1
4
.2
6
.3
8
.5
6) Explain why the following is or is not a valid probability distribution for the discrete random variable x.
x
p(x)
10
.3
20
.2
30
.2
40
.2
50
.2
7) Consider the given discrete probability distribution. Construct a graph for p(x).
x
p(x)
1
.1
2
.2
3
.2
4
.3
5
.2
8) Consider the given discrete probability distribution. Construct a graph for p(x).
x
p(x)
1
.30
2
.25
3
.20
4
.15
5
.05
6
.05
2 Find Probability Given Distribution
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Consider the given discrete probability distribution. Find the probability that x equals 5.
x
3
P(x) 0.33
A) 0.09
5
?
x
2
P(x) 0.28
A) 0.44
5
?
7
0.27
9
0.31
B) 0.91
C) 0.45
D) 4.55
B) 0.72
C) 0.56
D) 0.28
C) .2
D) .3
2) Consider the given discrete probability distribution. Find the probability that x exceeds 5.
6
0.25
8
0.19
3) Consider the given discrete probability distribution. Find P(x
x
p(x)
A) .5
1
.1
2
.2
3
.2
B) .7
4
.3
5
.2
3).
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4) Consider the given discrete probability distribution. Find P(x
x
p(x)
A) .95
0
.30
1
.25
2
.20
B) .90
3
.15
4
.05
4).
5
.05
C) .05
D) .10
C) 0.30
D) 0.60
5) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is
shown below. Find the probability that the random variable x is a value greater than 5.
x
2
3
5
8
10
p(x) 0.10 0.20 0.30 0.30 0.10
A) 0.40
B) 0.70
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) Consider the given discrete probability distribution. Find P(x
2 or x
3).
7) Consider the given discrete probability distribution. Find P(x
1 or x
2).
1
.1
x
p(x)
x
p(x)
0
.30
2
.2
1
.25
3
.2
2
.20
4
.3
3
.15
5
.2
4
.05
5
.05
4.3 Expected Values of Discrete Random Variables
1 Find Expected Value
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution
below:
Price
Probability
$10.00
0.4
$12.50
0.45
$15.00
0.15
How much should the lab budget for next year's frog orders assuming this distribution does not change? (Hint:
Find the expected price and assume 52 weeks per year.)
A) $617.50
B) $11.88
C) $1188.00
D) $3,211,000.00
2) Mamma Temte bakes six pies each day at a cost of $2 each. On 39% of the days she sells only two pies. On 38%
of the days, she sells 4 pies, and on the remaining 23% of the days, she sells all six pies. If Mama Temte sells her
pies for $4 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given
away.]
A) $2.72
B) $14.72
C) $8.00
D) $8.32
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3) A local bakery has determined a probability distribution for the number of cheesecakes it sells in a given day.
The distribution is as follows:
Number sold in a day
Prob (Number sold)
0
0.14
5
0.16
10
0.23
15
0.17
20
0.3
Find the number of cheesecakes that this local bakery expects to sell in a day.
A) 11.65
B) 11.79
C) 20
D) 10
4) A dice game involves rolling three dice and betting on one of the six numbers that are on the dice. The game
costs $11 to play, and you win if the number you bet appears on any of the dice. The distribution for the
outcomes of the game (including the profit) is shown below:
Number of dice with your number
0
1
2
3
Profit
$11
$11
$13
$33
Find your expected profit from playing this game.
A) $1.53
B) $0.50
Probability
125/216
75/216
15/216
1/216
C) $11.20
D) $5.96
Answer the question True or False.
5) The expected value of a discrete random variable must be one of the values in which the random variable can
result.
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
6) An airline has requests for standby flights at half of the usual one way air fare. Past experience has shown that
these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as
a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one way air
fare to a certain city is $156 and the cost of flying first class is $355. Should a passenger who wishes to fly to this
city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.]
7) An automobile insurance company estimates the following loss probabilities for the next year on a $25,000
sports car:
Total loss:
50% loss:
25% loss:
10% loss:
No loss:
0.001
0.01
0.05
0.10
0.839
Assuming the company will sell only a $500 deductible policy for this model (i.e., the owner covers the first
$500 damage), how much annual premium should the company charge in order to average $620 profit per
policy sold?
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2 Find Mean, Variance, Standard Deviation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Calculate the mean for the discrete probability distribution shown here.
X
1
4
8
11
P(X) 0.28 0.24 0.17 0.31
A) 6.01
B) 6
C) 24
D) 1.5025
C) 5.6
D) 5.5
C) 5.7
D) 1.845
2) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is
shown below. Find the mean of the distribution.
x
2
3
5
8
10
p(x) 0.10 0.20 0.30 0.30 0.10
A) 5.7
B) 5.0
3) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is
shown below. Find the standard deviation of the distribution.
x
2
3
5
8
10
p(x) 0.10 0.20 0.30 0.30 0.10
A) 6.41
B) 2.532
4) A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution
below:
Price
Probability
$10.00
0.25
$12.50
0.45
$15.00
0.3
Suppose the mean cost of the frogs is $12.63 per week. Interpret this value.
A) The average cost for all weekly frog purchases is $12.63.
B) Most of the weeks resulted in frog costs of $12.63.
C) The median cost for the distribution of frog costs is $12.63.
D) The frog cost that occurs more often than any other is $12.63.
5) The random variable x represents the number of boys in a family with three children. Assuming that births of
boys and girls are equally likely, find the mean and standard deviation for the random variable x.
A) mean: 1.50; standard deviation: .87
B) mean: 2.25; standard deviation: .87
C) mean: 1.50; standard deviation: .76
D) mean: 2.25; standard deviation: .76
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6) In a pizza takeout restaurant, the following probability distribution was obtained for the number of toppings
ordered on a large pizza. Find the mean and standard deviation for the random variable.
x
P(x)
0
.30
1
.40
2
.20
3
.06
4
.04
A) mean: 1.14; standard deviation: 1.04
C) mean: 1.30; standard deviation: 2.38
B) mean: 1.54; standard deviation: 1.30
D) mean: 1.30; standard deviation: 1.54
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
7) Find the mean and standard deviation of the probability distribution for the random variable x, which
represents the number of cars per household in a small town.
x
0
1
2
3
4
P(x)
.125
.428
.256
.108
.083
8) Calculate the mean for the discrete probability distribution shown here.
X
2 3 8 10
P(X) .2 .3 .3 .2
9) Consider the given discrete probability distribution.
a.
x
p(x)
Find
1
.1
E(x).
2
.2
3
.2
4
.3
5
.2
b. Find
E[(x
)2 ].
c. Find the probability that the value of x falls within one standard deviation of the mean. Compare this result
to the Empirical Rule.
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4.4 The Binomial Random Variable
1 Understand the Binomial Random Variable
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A recent article in the paper claims that business ethics are at an all time low. Reporting on a recent sample, the
paper claims that 42% of all employees believe their company president possesses low ethical standards.
Suppose 20 of a company's employees are randomly and independently sampled and asked if they believe
their company president has low ethical standards and their years of experience at the company. Could the
probability distribution for the number of years of experience be modelled by a binomial probability
distribution?
A) No, a binomial distribution requires only two possible outcomes for each experimental unit sampled.
B) Yes, the sample is a random and independent sample.
C) Yes, the sample size is n 20.
D) No, the employees would not be considered independent in the present sample.
2) Which binomial probability is represented on the screen below?
A) The probability of 2 successes in 8 trials where the probability of success is .3.
B) The probability of 8 failures in 2 trials where the probability of failure is .3.
C) The probability of 2 successes in 8 trials where the probability of failure is .3.
D) The probability of 8 successes in 2 trials where the probability of success is .3.
3) Which binomial probability is represented on the screen below?
A) P(x
4)
B) P(x
4)
C) P(x
4)
D) P(x
4)
4) For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the
probability of success is .4?
A) the probability of 4 failures in 5 trials where the probability of success is .6
B) the probability of 1 success in 5 trials where the probability of failure is .6
C) the probability of 4 failures in 5 trials where the probability of success is .4
D) the probability of 4 failures in 5 trials where the probability of failure is .6
5) Compute
A) 35
7!
.
3!(7 3)!
B) 210
C) 840
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D) 70
6) Compute
9
.
4
7) Compute
5
.
0
8) Compute
4
.
4
9) Compute
5
.
4
A) 126
A) 1
A) 1
A) 5
B) 84
C) 3024
D) 15,120
B) 5
C) 10
D) undefined
B) 4
C) 6
D) 16
B) 1
C) 10
D) 20
10) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20
eligible voters were randomly selected from the population of all eligible voters. Which of the following is
necessary for this problem to be analyzed using the binomial random variable?
I. There are two outcomes possible for each of the 20 voters sampled.
II. The outcomes of the 20 voters must be considered independent of one another.
III. The probability a voter will actually vote is 0.70, the probability they won't is 0.30.
A) I only
B) II only
C) III only
D) I, II, and III
Answer the question True or False.
11) A binomial random variable is defined to be the number of units sampled until x successes is observed.
A) True
B) False
12) The binomial distribution can be used to model the number of rare events that occur over a given time period.
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
13) Compute
6
(.3) 2 (.7)6 2 .
2
14) For a binomial distribution, if the probability of success is .63 on the first trial, what is the probability of success
on the second trial?
15) For a binomial distribution, if the probability of success is .48 on the first trial, what is the probability of failure
on the second trial?
2 Find Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem. Round to four decimal places.
1) If x is a binomial random variable, compute p(x) for n
A) 0.2592
B) 0.2411
2) If x is a binomial random variable, compute p(x) for n
A) 0.4096
B) 0.0064
5, x
5, x
1, p 0.4.
C) 0.2929
1, q 0.8.
C) 0.3850
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D) 0.2722
D) 0.0068
Solve the problem.
3) According to a recent study, 1 in every 9 women has been a victim of domestic abuse at some point in her life.
Suppose we have randomly and independently sampled twenty five women and asked each whether she has
been a victim of domestic abuse at some point in her life. Find the probability that at least 2 of the women
sampled have been the victim of domestic abuse. Round to six decimal places.
A) 0.782924
B) 0.246677
C) 0.536248
D) 0.217076
4) According to a recent study, 1 in every 6 women has been a victim of domestic abuse at some point in her life.
Suppose we have randomly and independently sampled twenty five women and asked each whether she has
been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women
sampled have not been the victim of domestic abuse.
A) 0.188687
B) 0.062896
C) 0.125791
D) 0.807120
5) We believe that 90% of the population of all Business Statistics students consider statistics to be an exciting
subject. Suppose we randomly and independently selected 24 students from the population and observed
fewer than five in our sample who consider statistics to be an exciting subject. Make an inference about the
belief that 90% of the students consider statistics to be an exciting subject.
A) The 90% number is too high. The real percentage is lower than 90%.
B) The 90% number is too low. The real percentage is higher than 90%.
C) The 90% number is exactly right.
D) It is impossible to make any inferences about the 90% number based on this information.
6) We believe that 82% of the population of all Business Statistics students consider statistics to be an exciting
subject. Suppose we randomly and independently selected 39 students from the population. If the true
percentage is really 82%, find the probability of observing 38 or more students who consider statistics to be an
exciting subject. Round to six decimal places.
A) 0.004161
B) 0.000435
C) 0.003726
D) 0.995839
7) A literature professor decides to give a 15 question true false quiz. She wants to choose the passing grade such
that the probability of passing a student who guesses on every question is less than .10. What score should be
set as the lowest passing grade?
A) 11
B) 9
C) 12
D) 10
8) A recent article in the paper claims that business ethics are at an all time low. Reporting on a recent sample, the
paper claims that 42% of all employees believe their company president possesses low ethical standards.
Assume that responses were randomly and independently collected. A president of a local company that
employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many
of her company's employees believe that she possesses low ethical standards?
A) 420
B) 42
C) 580
D) 958
9) A recent article in the paper claims that business ethics are at an all time low. Reporting on a recent sample, the
paper claims that 44% of all employees believe their company president possesses low ethical standards.
Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim
is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's
president possesses low ethical standards. Round to six decimal places.
A) 0.437608
B) 0.285201
C) 0.669843
D) 0.809834
10) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20
eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability
table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential
election.
A) 0.392
B) 0.228
C) 0.772
D) 0.608
E) 0.887
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11) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20
eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability
table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in
the next presidential election.
A) 0.780
B) 0.714
C) 0.845
D) 0.649
12) It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small
private college decided to randomly and independently sample 15 of their students and ask if they have a
tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they
did have at least one tattoo.
A) 0.722
B) 0.515
C) 0.207
D) 0.218
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
13) About 40% of the general population donate time and energy to community projects. Suppose 15 people have
been randomly selected from a community and each asked whether he or she donates time and energy to
community projects. Let x be the number who donate time and energy to community projects. Use a binomial
probability table to find the probability that more than five of the 15 donate time and energy to community
projects.
14) An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact
model are defective. If 14 of these cars are independently sampled, what is the probability that more than half
need new gas tanks?
15) A new drug is designed to reduce a person's blood pressure. Thirteen randomly selected hypertensive patients
receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is
untreated is 0.5. Then what is the probability of observing 11 or more blood pressure drops in a random sample
of 13 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal
places.
16) A local newspaper claims that 70% of the items advertised in its classifieds section are sold within 1 week of the
first appearance of the ad. To check the validity of the claim, the newspaper randomly selected n 25
advertisements from last year's classifieds and contacted the people who placed the ads. They found that 16 of
the 25 items sold within a week. Based on the newspaper's claim, is it likely to observe x 16 who sold their
item within a week? Use a binomial probability table.
3 Find Mean/Expected Value, Standard Deviation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If x is a binomial random variable, calculate
A) 28
B) 35
for n
2) If x is a binomial random variable, calculate 2 for n
A) 11.2
B) 14
3) If x is a binomial random variable, calculate
necessary.
A) 4.583
B) 30
for n
70 and p 0.4.
C) 2.8
70 and p 0.2.
C) 3.347
100 and p
D) 16.8
D) 2.8
0.3. Round to three decimal places when
C) 21
D) 5.477
4) The probability that an individual is left handed is 0.16. In a class of 80 students, what is the mean and
standard deviation of the number of left handed students? Round to the nearest hundredth when necessary.
A) mean: 12.8; standard deviation: 3.28
B) mean: 80; standard deviation: 3.28
C) mean: 12.8; standard deviation: 3.58
D) mean: 80; standard deviation: 3.58
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5) A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 20 adults
over 50, what is the mean and standard deviation of the number who wear glasses? Round to the nearest
hundredth when necessary.
A) mean: 12.6; standard deviation: 2.16
B) mean: 12.6; standard deviation: 3.55
C) mean: 7.4; standard deviation: 2.16
D) mean: 7.4; standard deviation: 3.55
6) According to a published study, 1 in every 4 men has been involved in a minor traffic accident. Suppose we
have randomly and independently sampled twenty five men and asked each whether he has been involved in
a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic
accident? Round to the nearest whole number.
A) 19
B) 4
C) 6
D) 25
7) We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting
subject. Suppose we randomly and independently selected 39 students from the population. How many of the
sampled students do we expect to consider statistics to be an exciting subject?
A) 31.59
B) 33.82
C) 39
D) 32.16
8) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20
eligible voters were randomly selected from the population of all eligible voters. How many of the sampled
voters do we expect to vote in the next presidential election?
A) 0.7
B) 0.3
C) 14
D) 6
9) It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small
private college decided to randomly and independently sample 15 of their students and ask if they have a
tattoo. Find the standard deviation for this binomial random variable. Round to the nearest hundredth when
necessary.
A) 4.5
B) 10.5
C) 3.15
D) 1.77
4.5 The Poisson Random Variable (Optional)
1 Find Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The number of road construction projects that take place at any one time in a certain city follows a Poisson
distribution with a mean of 6. Find the probability that exactly four road construction projects are currently
taking place in this city.
A) 0.133853
B) 0.032968
C) 0.423040
D) 0.104196
2) The number of road construction projects that take place at any one time in a certain city follows a Poisson
distribution with a mean of 7. Find the probability that more than four road construction projects are currently
taking place in the city.
A) 0.827008
B) 0.918235
C) 0.172992
D) 0.081765
3) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson
distribution with a mean of 7.7. Find the probability that fewer than three accidents will occur next month on
this stretch of road.
A) 0.017364
B) 0.051819
C) 0.982636
D) 0.948181
4) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson
distribution with a mean of 7.1. Find the probability of observing exactly four accidents on this stretch of road
next month.
A) 0.087364
B) 1.939297
C) 0.647029
D) 14.362723
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5) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson
distribution with a mean of 7.2. Find the probability that exactly five accidents will occur on this stretch of road
each of the next two months.
A) 0.014492
B) 0.120382
C) 0.240764
D) 0.002876
6) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 5.
Find the probability that exactly six babies will be born during a particular 1 hour period at this hospital.
A) 0.146223
B) 0.000044
C) 0.018278
D) 0.000031
7) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 6.
Some people believe that the presence of a full moon increases the number of births that take place. Suppose
during the presence of a full moon, the hospital experienced eight consecutive hours with more than seven
births each hour. Based on this fact, comment on the belief that the full moon increases the number of births.
A) The belief is supported as the probability of observing this many births would be 0.0000184.
B) The belief is not supported as the probability of observing this many births is 0.256.
C) The belief is supported as the probability of observing this many births would be 0.256.
D) The belief is not supported as the probability of observing this many births is 0.0000184.
8) The university police department must write, on average, five tickets per day to keep department revenues at
budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of
7.4. Find the probability that fewer than six tickets are written on a randomly selected day.
A) 0.252557
B) 0.391962
C) 0.747443
D) 0.608038
9) The university police department must write, on average, five tickets per day to keep department revenues at
budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of
8.9. Find the probability that exactly four tickets are written on a randomly selected day.
A) .035656
B) .964344
C) .058433
D) .941567
10) The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean
of 4 goals per game. Find the probability that the team will score more than three goals during a game.
A) 0.566530
B) 0.433470
C) 0.761897
D) 0.238103
11) The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean
of 5 goals per game. Find the probability that the team scored exactly three goals in each of four randomly
selected games.
A) 0.00038828
B) 0.56149561
C) 0.00540243
D) 0.43850439
12) An alarm company reports that the number of alarms sent to their monitoring center from customers owning
their system follow a Poisson distribution with
4.6 alarms per year. Find the probability that a randomly
selected customer had more than 7 alarms reported.
A) 0.905
B) 0.095
C) 0.087
D) 0.182
E) 0.818
13) The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean
of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.
A) 0.895
B) 0.105
C) 0.219
D) 0.114
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) A small life insurance company has determined that on the average it receives 3 death claims per day. Find the
probability that the company receives at least seven death claims on a randomly selected day.
15) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson
distribution with a mean of 7.2. Find the probability that fewer than two accidents will occur on this stretch of
road during a month.
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16) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3.
Find the probability that exactly two babies are born during a randomly selected hour.
17) Compute
xe
x!
for
5 and x
7.
2 Find Mean/Expected Value, Standard Deviation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The university police department must write, on average, five tickets per day to keep department revenues at
budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of
7.5. Interpret the value of the mean.
A) If we sampled all days, the arithmetic average number of tickets written would be 7.5 tickets per day.
B) The number of tickets that is written most often is 7.5 tickets per day.
C) On half of the days less than 7.5 tickets are written and on half of the days have more than 7.5 tickets are
written.
D) The mean has no interpretation since 0.5 ticket can never be written.
2) Suppose a Poisson probability distribution with
random variable x.. Find for x.
A) 0.8
B) 0.8
3) Suppose a Poisson probability distribution with
random variable x. Find for x.
A) 9.6
B) 9.6
0.8 provides a good approximation of the distribution of a
C) 0.4
D) 0.64
C) 4.8
D) 92.16
9.6 provides a good approximation of the distribution of a
4) An alarm company reports that the number of alarms sent to their monitoring center from customers owning
their system follow a Poisson distribution with
4.7 alarms per year. Identify the mean and standard
deviation for this distribution.
A) mean 4.7, standard Deviation 4.7
B) mean 4.7, standard Deviation 2.17
C) mean 2.17, standard Deviation 4.7
D) mean 2.17, standard Deviation 2.17
5) The number of homeruns hit during a major league baseball game follows a Poisson distribution with
Find the mean and standard deviation for this distribution.
A) mean 3.2, standard Deviation 3.2
B) mean 3.2, standard Deviation 10.24
C) mean 3.2, standard Deviation 1.79
D) mean 1.79, standard Deviation 3.2
3.2.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) Suppose x is a random variable for which a Poisson probability distribution
with
3 provides a good approximation.
a.
b.
c.
Graph p(x) for x 0, 1, 2, 3, 4, 5, 6.
Find and for x.
What is the probability that x will fall in the interval
?
7) A bank offers online banking to its customers free of charge. While online, customers can also sign up for
additional services that the bank offers. Let x be the number of customers who sign up for additional services
online each day. Suppose the distribution of x is approximated well by a Poisson distribution with mean
42.3. Find E(x) and interpret its value.
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4.6 The Hypergeometric Random Variable (Optional)
1 Identify the Characteristics of a Hypergeometric Random Variable
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
1) The conditions for both the hypergeometric and the binomial random variables require that each trial results in
one of two outcomes.
A) True
B) False
2) The conditions for both the hypergeometric and the binomial random variables require that the trials are
independent.
A) True
B) False
3) The hypergeometric random variable x counts the number of successes in the draw of n elements from a set of
N elements containing r successes.
A) True
B) False
4) The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of
12 elements containing 7 successes. The numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all possible values of x.
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
5) The hypergeometric random variable x counts the number of successes in the draw of 3 elements from a set of
8 elements containing 4 successes. List the possible values of x.
6) The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of
10 elements containing 2 successes. List the possible values of x.
2 Find Probability
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Given that x is a hypergeometric random variable, compute p(x) for N
A) .45
B) .125
C) .375
2) Given that x is a hypergeometric random variable, compute p(x) for N
A) .536
B) .464
C) .140
3) Given that x is a hypergeometric random variable with N
A) .033
B) .200
4) Given that x is a hypergeometric random variable with N
A) 0
B) .001
6, n
3, r
8, n
5, r
10, n 3, and r
C) .216
15, n 6, and r
C) .002
3, and x 1.
D) .55
3, and x 2.
D) .343
6, compute P(x 0).
D) 0
10, compute P(x 0).
D) 1
5) Suppose that 4 out of 12 liver transplants done at a hospital will fail within a year. Consider a random sample
of 3 of these 12 patients. What is the probability that all 3 patients will result in failed transplants?
A) .018
B) .037
C) .296
D) .333
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6) Suppose the candidate pool for two appointed positions includes 6 women and 9 men. All candidates were told
that the positions were randomly filled. Find the probability that two men are selected to fill the appointed
positions.
A) .343
B) .143
C) .160
D) .360
7) As part of a promotion, both you and your roommate are given free cellular phones from a batch of 13 phones.
Unknown to you, four of the phones are faulty and do not work. Find the probability that one of the two
phones is faulty.
A) .462
B) .538
C) .231
D) .077
8) Suppose a man has ordered twelve 1 gallon paint cans of a particular color (lilac) from the local paint store in
order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint.
For this weekend's big project, the man randomly selects four of these 1 gallon cans to paint his mother's living
room. Let x the number of the paint cans selected that are defective. Unknown to the man, x follows a
hypergeometric distribution. Find the probability that none of the four cans selected contains an incorrect mix
of paint.
A) 0.50909
B) 0.21818
C) 0.01818
D) 0.25455
9) Suppose a man has ordered twelve 1 gallon paint cans of a particular color (lilac) from the local paint store in
order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint.
For this weekend's big project, the man randomly selects four of these 1 gallon cans to paint his mother's living
room. Let x the number of the paint cans selected that are defective. Unknown to the man, x follows a
hypergeometric distribution. Find the probability that at least one of the four cans selected contains an incorrect
mix of paint.
A) 0.50909
B) 0.78182
C) 0.49091
D) 0.74545
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10) Given that x is a hypergeometric random variable with N
10, n
11) Given that x is a hypergeometric random variable with N
8, n
a.
b.
c.
d.
P(x 0)
P(x 1)
P(x 1)
P(x 2)
a. Display the probability distribution in tabular form.
b. Find P(x 2).
5, and r
4, and r
6, find each probability.
3:
12) You test 3 items from a lot of 12. What is the probability that you will test no defective items if the lot contains
2 defective items?
13) You test 4 items from a lot of 15. What is the probability that you will test no defective items if the lot contains
3 defective items?
14) You randomly select 7 students from a class with 15 male and 20 female students. What is the probability that
you will choose exactly 4 females?
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3 Find Mean, Variance, Standard Deviation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Given that x is a hypergeometric random variable with N
A) 3
B) 1
2) Given that x is a hypergeometric random variable with N
A) .538
B) .469
3) Given that x is a hypergeometric random variable with N
of x.
A) .745
B) .556
10, n 5, and r
C) 2
8, n 4, and r
C) .700
9, n
3, and r
C) .208
6, compute the mean of x.
D) 4
3, compute the variance of x.
D) .732
5, compute the standard deviation
D) .456
4) Suppose a man has ordered twelve 1 gallon paint cans of a particular color (lilac) from the local paint store in
order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint.
For this weekend's big project, the man randomly selects four of these 1 gallon cans to paint his mother's living
room. Let x the number of the paint cans selected that are defective. Unknown to the man, x follows a
hypergeometric distribution. Find the mean of this distribution.
A) 4
B) 1
C) 3
D) 12
5) Suppose a man has ordered twelve 1 gallon paint cans of a particular color (lilac) from the local paint store in
order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint.
For this weekend's big project, the man randomly selects four of these 1 gallon cans to paint his mother's living
room. Let x the number of the paint cans selected that are defective. Unknown to the man, x follows a
hypergeometric distribution. Find the standard deviation of this distribution.
A) 0.739
B) 0.545
C) 1
D) 0.297
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6) Given that x is a hypergeometric random variable with N
a. Display the probability distribution in tabular form.
b. Compute and for x.
c. What is the probability that x will fall within the interval
10, n
3, and r
2
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