Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1)
60% 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 (0.6)(0.6)(0.6)(0.6) =0.130
1)
Provide an appropriate response.
2)
Give an example of a discrete random variable whose possible values form a countable
infinite set of numbers.
2)
Solve the problem.
3)
12% 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 (0.12)(0.12)(0.12) =0.002
3)
Provide an appropriate response.
4)
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.
4)
5)
An experiment consists of randomly selecting a card from a deck of 52. The event A is
defined as follows.
A = event the card selected is a diamond
Give an example of a pair of events B and C for this experiment such that the events A and
B are mutually exclusive but the collection of events A, B, and C is not mutually exclusive.
5)
6)
Suppose that you roll a die and record the number that comes up and then flip a coin and
record whether it comes up heads or tails. One possible outcome can be represented as 2H
(a two on the die followed by heads). Make a list of all the possible outcomes. What is the
probability that you get tails and an even number? What assumption are you making
when you find this probability?
6)
7)
Describe an event whose probability of occurring is 1 and explain what that probability
means. Describe an event whose probability of occurring is 0 and explain what that
probability means.
7)
8)
Explain in your own words the meaning of the term “probability distribution”.
8)
9)
Suppose that in an election for governor of Oregon there are five candidates of whom two
are women. A statistics student reasons as follows. The probability that a woman will win
the election is equal to f
N which is 2
5. What is wrong with his reasoning?
9)
10)
Discuss the range of possible values for probabilities. Give examples to support each.
10)
11)
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?
11)
12)
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.
12)
13)
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?
13)
14)
Construct a Venn diagram portraying four events A, B, C, and D such that the collection of
events A, B, and C is mutually exclusive, the collection of events A, B, and D is mutually
exclusive, but the collection of events A, B, C, and D is not mutually exclusive.
14)
15)
On an exam question asking for a probability, Sue had an answer of 13
8. Explain how she
knew that this result was incorrect.
15)
16)
Interpret the following probability statement using the frequentist interpretation of
probability. The probability is 0.83 that this particular type of surgery will be successful.
16)
17)
Construct a Venn diagram representing the event ((not A) or B).
17)
18)
Construct a Venn diagram representing the event ((A & B) or C).
18)
19)
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?
19)
20)
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.
20)
21)
Explain how you would construct a probability histogram of a discrete random variable
given its probability distribution.
21)
22)
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.
22)
23)
An experiment consists of randomly selecting a card from a deck of 52. The event A is
defined as follows.
A = event the card selected is a diamond
Give an example of an event B for this experiment such that the events A and B are
mutually exclusive.
23)
24)
Construct a Venn diagram portraying three events A, B, and C such that A and B are
mutually exclusive, B and C are mutually exclusive, but the collection of events A, B, and C
is not mutually exclusive.
24)
25)
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.
25)
26)
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.
26)
Solve the problem.
27)
A coin is biased so that the probability it will come up tails is 0.43. 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 (0.57)(0.57)(0.57) =0.185
27)
Provide an appropriate response.
28)
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?
28)
29)
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.
29)
30)
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
30)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculate the specified probability
31)
31)
A)
1.125
B)
0.2875
C)
0.425
D)
0.575
Obtain the probability distribution of the random variable.
32)
32)
A)
Siblings
xProbability
P(X = x)
00.300
10.366
20.185
30.086
40.038
50.015
60.009
70.002
B)
Siblings
xProbability
P(X = x)
00.125
10.125
20.125
30.125
40.125
50.125
60.125
70.125
C)
Siblings
xProbability
P(X = x)
00.315
10.351
20.197
30.074
40.038
50.015
60.009
70.002
D)
Siblings
xProbability
P(X = x)
10.524
20.264
30.122
40.054
50.021
60.013
70.002
Provide an appropriate response.
33)
33)
A)
p =8
15, n = 3
B)
p =8
15, n =15
C)
p =1
3, n = 3
D)
p =1
8, n =15
Find the indicated probability by using the complementation rule.
34)
34)
A)
0.093
B)
0.915
C)
0.917
D)
0.085
Find the indicated probability.
35)
35)
A)
5
9
B)
7
9
C)
1
3
D)
4
5
Find the indicated probability by using the complementation rule.
36)
36)
A)
0.866
B)
0.196
C)
0.134
D)
0.330
Determine whether the events are mutually exclusive.
37)
37)
A)
Yes
B)
No
Find the indicated probability.
38)
38)
A)
2
10
B)
1
10
C)
3
10
D)
2
5
Find the indicated probability. Round to four decimal places.
39)
39)
A)
0.4256
B)
0.2254
C)
0.3041
D)
0.2001
Find the indicated probability.
40)
40)
A)
1
142
B)
27
71
C)
44
71
D)
44
27
Find the indicated probability by using the complementation rule.
41)
41)
A)
0.22
B)
2.27
C)
0.79
D)
0.56
Find the mean of the binomial random variable. Round to two decimal places when necessary.
42)
42)
A)
4.4
B)
4
C)
16
D)
17.6
Find the indicated probability.
43)
43)
A)
1
12
B)
1
9
C)
1
36
D)
1
18
Provide an appropriate response.
44)
44)
A)
True
B)
False
Find the indicated binomial probability. Round to five decimal places when necessary.
45)
45)
A)
0.30199
B)
0.00007
C)
1.8
D)
0.00671
Estimate the probability of the event.
46)
46)
A)
0.8
B)
0.6
C)
0.4
D)
0.7
Find the standard deviation of the binomial random variable.
47)
47)
A)
1.568
B)
1.652
C)
1.632
D)
1.587
Estimate the probability of the event.
48)
48)
A)
0.261
B)
0.118
C)
0.114
D)
0.171
List the outcomes comprising the specified event.
49)
49)
A)
JH, GH, HJ, HG, HM
B)
JH, GH, HJ, JG, HG, HM, MH
C)
JH, GH, HJ, HG, HM, MH
D)
HJ, HG, HM
Find the expected value of the random variable. Round to the nearest cent unless stated otherwise.
50)
50)
A)
$12.13
B)
$2.47
C)
–$9.67
D)
–$7.20
Evaluate the expression.
51)
51)
A)
440
B)
4
C)
220
D)
362,880
Find the indicated probability by using the complementation rule.
52)
52)
A)
0.658
B)
0.724
C)
0.276
D)
0.278
53)
53)
A)
0.341
B)
0.211
C)
0.659
D)
0.552
Find the indicated binomial probability. Round to five decimal places when necessary.
54)
54)
A)
2.6665
B)
3453.87152
C)
0.22105
D)
2.88
Describe the specified event in words.
55)
55)
A)
The event the student took less than 10 hours or more than 6 hours
B)
The event the student took less than 10 hours and between 6 and 6 hours inclusive
C)
The event the student took less than 10 hours or between 6 and 6 hours inclusive
D)
The event the student took between 10 and 6 hours inclusive
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.
56)
56)
A)
0.037
B)
0.048
C)
0.600
D)
0.055
Find the indicated binomial probability. Round to five decimal places when necessary.
57)
57)
A)
0
B)
144.9646
C)
0.14546
D)
104.72847
Find the specified probability.
58)
58)
A)
0.40
B)
0.20
C)
0.95
D)
0.10
Find the indicated probability.
59)
59)
A)
1
8
B)
3
8
C)
5
8
D)
1
2
Find the indicated binomial probability. Round to five decimal places when necessary.
60)
60)
A)
0.32813
B)
0.65625
C)
0.00781
D)
0.16406
Describe the specified event in words.
61)
61)
A)
The event the student took more than 7 hours and less than 9 hours
B)
The event the student took between 7 and 9 hours inclusive
C)
The event the student took between 5 and 7 hours inclusive
D)
The event the student at least 5 hours
Find the indicated probability by using the general addition rule.
62)
62)
A)
0.690
B)
0.548
C)
0.599
D)
0.497
Find the specified probability.
63)
63)
A)
0.15
B)
0.55
C)
0.05
D)
0.10
Evaluate the expression.
64)
64)
A)
28
B)
29
C)
1
D)
30
65)
65)
A)
2!
B)
63,000
C)
72
D)
9
7
Find the indicated probability by using the general addition rule.
66)
66)
A)
9
B)
9
19
C)
19
4
D)
4
19
Find the specified probability distribution of the binomial random variable.
67)
67)
A)
x P(X = x)
00.0715
10.2614
20.3452
30.2398
40.0705
50.0116
B)
x P(X = x)
10.41
20.1681
30.0689
40.0283
50.0116
C)
x P(X = x)
00.0715
10.2484
20.3232
30.2618
40.0835
50.0116
D)
x P(X = x)
00.0715
10.2484
20.3452
30.2398
40.0835
50.0116
Find the indicated probability by using the complementation rule.
68)
68)
A)
0.985
B)
0.044
C)
0.029
D)
0.956
Determine the possible values of the random variable.
69)
69)
A)
2, 4, 6, 15
B)
2, 4, 6, 5
C)
2, 4, 6, 15, 5
D)
32
Find the indicated probability.
70)
70)
A)
1
365
B)
1
31
C)
31
365
D)
1
12
Provide an appropriate response.
71)
71)
A)
True
B)
False
Find the standard deviation of the binomial random variable.
72)
72)
A)
0.833
B)
0.818
C)
0.849
D)
0.836
List the outcomes comprising the specified event.
73)
73)
A)
JG, HG, MG, GM
B)
JG, HG
C)
JG, HG, MG
D)
JG, HG, MG, GJ, GH, GM
Determine the number of outcomes that comprise the specified event.
74)
74)
A)
79
B)
97
C)
111
D)
10
Find the indicated probability by using the general addition rule.
75)
75)
A)
0.655
B)
0.057
C)
0.068
D)
0.622
List the outcomes comprising the specified event.
76)
76)
A)
ABD, ACD, ADE
B)
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BDE, CDE
C)
ABC, ABE, ACE, BCD, BDE, CDE
D)
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BDE
B
Use random–variable notation to represent the event.
77)
77)
A)
P{X = 3}
B)
{X
3}
C)
{X = 3}
D)
HTTT, THTT, TTHT, TTTH
C
D
Find the standard deviation of the binomial random variable.
78)
78)
A)
2.6609
B)
0.1143
C)
0.0109
D)
0.1044
Describe the specified event in words.
79)
79)
A)
The event the student is between 21 and 35 inclusive
B)
The event the student is between 24 and 35 inclusive
C)
The event the student is 21 or over
D)
The event the student is between 35 and 40 inclusive
Find the mean of the random variable.
80)
80)
A)
0.70
B)
1.21
C)
2.00
D)
0.80
Calculate the specified probability
81)
81)
A)
0
B)
0.975
C)
0.0125
D)
0.025
Find the indicated probability. Round to four decimal places.
82)
82)
A)
0.3125
B)
0.1875
C)
0.5000
D)
0.8125
List the outcomes comprising the specified event.
83)
83)
A)
ABC, ABD, ACE, ADE, BCE, BDE
B)
ABC, ABD, ADE, BCE, BDE
C)
ACD, BCD, CDE
D)
ABC, ABD, ABE, ACE, ADE, BCE, BDE
Draw a Venn diagram and shade the described events.
84)
84)
A)
B)
C)
D)
Describe the specified event in words.
85)
85)
A)
Event that exactly two tails are tossed or the first toss is heads or both
B)
Event that exactly two tails are tossed and the first toss is heads
C)
Event that exactly two tails are tossed or the first toss is heads but not both
D)
Event that the first toss is heads or the last two tosses are tails or both
Provide an appropriate response.
86)
86)
A)
True
B)
False
Describe the specified event in words.
87)
87)
A)
The event that exactly two tails are tossed or the last two tosses are heads or both
B)
The event that two of the tosses come up tails and two tosses come up heads
C)
The event the first two tosses are tails and the last two tosses are heads
D)
The event that the first two tosses come up the same and the last two tosses come up the same
Determine the number of outcomes that comprise the specified event.
88)
88)
A)
34
B)
36
C)
8
D)
70
Construct the requested histogram.
89)
89)
A)
B)
C)
D)
Find the indicated probability by using the general addition rule.
90)
90)
A)
101
264
B)
3
88
C)
1
24
D)
35
264
Use random–variable notation to represent the event.
91)
91)
A)
{0, 1}
B)
{X < 2}
C)
P{X < 2}
D)
{X
2}
Determine the possible values of the random variable.
92)
92)
A)
0, 1, 2, 3, 4, 5, 6, 7
B)
189, 245, 102, 42, 24, 13, 5, 2
C)
Brother, sister
D)
7
A
93)
93)
A)
HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT,
TTHH, TTHT, TTTH, TTTT
B)
0, 1, 2, 3, 4
C)
1, 2, 3, 4
D)
1, 2, 3
B
B
Use random–variable notation to represent the event.
94)
94)
A)
(5, 6), (6, 5), (6,6)
B)
{Y
11}
C)
{Y > 11}
D)
{X+Y
11}
95)
95)
A)
{X = 2}
B)
X = 2
C)
{(1, 3), (2, 4), (3, 5), (4, 6), (3, 1), (4, 2), (5, 3), (6, 4)}
D)
P{X = 2}
A
Find the mean of the binomial random variable. Round to two decimal places when necessary.
96)
96)
A)
10.67
B)
0.33
C)
0.34
D)
0.32
B
List the outcome(s) of the stated event.
97)
97)
A)
Horses #4 and #6
B)
Horses #1 and #2
C)
Horses #1 and #3
D)
Horse #1
A
B
Find the indicated probability by using the special addition rule.
98)
98)
A)
0.302
B)
0.221
C)
0.698
D)
0.477
List the outcomes comprising the specified event.
99)
99)
A)
HJ, HG, HM, MH
B)
HM
C)
HJ, HG, HM
D)
HM, MH
Find the mean of the binomial random variable. Round to two decimal places when necessary.
100)
100)
A)
30
B)
70
C)
36
D)
84
Answer Key
Testname: C5
Answer Key
Testname: C5
35
Answer Key
Testname: C5
Answer Key
Testname: C5