Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
If X0=0.5
0.5 is an initial state vector for the transition matrix T=0.4 0.8
0.6 0.2 , compute the state vector
X2.
1)
A)
0.65
0.35
B)
0.34
0.66
C)
0.775
0.225
D)
0.56
0.44
E)
0.6
0.4
2)
For the matrix T=
11
21
4
01
41
4
01
41
2
, find the probability of going from state 2 to state 1 after two steps.
2)
A)
3
16
B)
11
16
C)
1
4
D)
1
8
E)
5
16
3)
A random variable X has a distribution given by f(1) = 0.6, f(2) = 0.2, f(3) = 0.2. The mean of X is
3)
A)
3.
B)
0.33.
C)
0.4.
D)
1.6.
E)
1.8.
4)
Suppose T=
1
21
4
1
23
4
is the transition matrix for a Markov chain. If Q=q1
q2 is the steady–state
vector, find q1.
4)
A)
1
6
B)
1
5
C)
1
2
D)
1
3
E)
1
4
5)
A manufacturer produces thermostats, 10% of which are defective. From a production run, a
sample of four thermostats is randomly selected. Determine the probability that at least three of the
selected thermostats are defective. You may assume that the four trials are independent and that
the number of defective items in the sample has a binomial distribution.
5)
A)
0.0038
B)
0.0036
C)
0.0037
D)
0.0035
E)
0.0039
6)
A biased coin is tossed 8 times. If the probability of heads appearing on any toss is 1
3, then the
probability that exactly six heads appear is
6)
A)
224
38.
B)
2
38.
C)
56
38.
D)
112
38.
E)
100
38.
7)
A random variance X has a distribution given by f(0) = 0.4, f(1) = 0.3, f(2) = 0.3. The variance of X is
7)
A)
0.50.
B)
0.69.
C)
0.47.
D)
0.32.
E)
0.54.
8)
If X is a binomial random variable involved with six independent trials, and the probability of
success on any trial is 1
4, then the probability that X= 2 is
8)
A)
2
46.
B)
9
46.
C)
1215
46.
D)
2430
46.
E)
81
46.
9)
Suppose that you pay $1 to play a game in which a pair of fair dice are rolled. For each 2 that shows
on a die you receive $2. To the nearest cent, your expected loss on each play is
9)
A)
$0.67.
B)
$0.33.
C)
$0.15.
D)
$0.07.
E)
$0.25.
10)
If a0.8
a+b a –b is a transitional matrix for a Markov chain, determine the value of b.
10)
A)
0.4
B)
0.8
C)
0.2
D)
0.6
E)
0.1
11)
From a group of 10 men and 10 women, two people are randomly selected to form a committee. If
X is the number of women on the committee, then P(X= 2) =
11)
A)
7
380.
B)
17
380.
C)
5
38.
D)
9
38.
E)
1
5.
12)
An urn contains 20 marbles, each of which shows a number. Eight marbles show 1, five marbles
show 2, and seven marbles show 3. A marble is randomly selected and the number X that shows is
observed. The mean of X is
12)
A)
39
20.
B)
3
10.
C)
1.
D)
23
20.
E)
3
2.
13)
A college dining hall has available two fruit juices for breakfast: orange and grapefruit. One
hundred students drink juice on a daily basis. It is found that a student will not drink grapefruit
juice on two successive days, and if a student drinks orange juice one day, then the following day
the student is equally likely to drink orange juice as to drink grapefruit juice. If 40 of the regular
juice drinkers drink orange juice on Monday, how many can be expected to drink orange juice on
Wednesday?
13)
A)
90
B)
50
C)
60
D)
10
E)
30
14)
An insurance company offers a life policy to individuals in a certain group. For a given 1–year
period, the company will pay $5000 in the event of the death of a policyholder. If the probability
that a person in this group dies during a 1–year time period is 0.001 and the annual premium for
the policy is $10, then the expected gain per policy to the company is
14)
A)
$5.00.
B)
$4.99.
C)
$4994.99.
D)
$9.90.
E)
$4990.
15)
A random variable X has a distribution given by f(0) = 0.3, f(1) = 0.6, f(2) = 0.1. (a) Find µ.
(b) Find .
15)
16)
Let X be the number of smokers who quit cold turkey. The probability of quitting cold
turkey is 10%. For a group of 5 smokers, selected independently, find the distribution of X.
16)
17)
A random variable X has a distribution given by f(0) = 0.3, f(1) = 0.2, f(2) = 0.5. (a) Find µ.
(b) Find Var(X).
17)
18)
Suppose that you pay $3 to play a game in which a fair die is rolled. You receive the
number of dollars equal to the number of dots that appear. What is your expected gain (or
loss) on each play?
18)
19)
A manufacturer produces light bulbs, of which 3% are defective. From a production run of
5000 bulbs, three bulbs are randomly selected and each is tested. Find, to three decimal
places, the probability that the sample contains exactly one defective bulb. Assume that the
three trials are independent and that the number of defective bulbs in the sample has a
binomial distribution.
19)
20)
If X0=0.1
0.9 is an initial state vector for the transition matrix T=0.3 0.5
0.7 0.5 , compute the
state vector X2.
20)
21)
For the transition matrix T=1 0.2
0 0.8 , find the probability of going from state 2 to state 1
after three steps.
21)
22)
If a c 0.1
b 0.2 0
0.4 c b +c is a transition matrix for a Markov chain, determine the values of a, b, and
c.
22)
23)
An urn contains three red and one green marble. Two marbles are randomly drawn in
succession without replacement. If X is the number of red marbles that are drawn and f is
the distribution of X, find f(0), f(1), and f(2).
23)
24)
Find the steady state vector for the transition matrix T=0.8 0.2
0.2 0.8 .
24)
25)
25% of a video store’s customers are female. Let X be the number of women among the
next 5 customers, and find the distribution of X.
25)
26)
Find the steady state vector for the transition matrix T=
11
3
02
3
.
26)
6
27)
If 0.2 0.6 0.4
0.3 a0.2
a +b0.1 0.2 is a transition matrix for a Markov chain, determine the value of b.
27)
28)
If X0=
1
2
1
2
is an initial state vector for the transition matrix T=
1
51
4
50, compute the state
vector X2.
28)
13
29)
For the transition matrix T=
1
21
4
1
23
4
, find the probability of going from state 1 to state 2
after two steps.
29)
30)
An urn contains 4 red and 6 green marbles. Two marbles are successively drawn at
random with replacement and the number of red marbles, X, is observed. Construct the
probability histogram for X.
30)
31)
If a person watches a certain TV daily evening news program on one evening, then the
probability that the person watches that program the next evening is 0.7. However, if the
person does not watch the program one evening, then the probability that the person
watches the program the next evening is 0.2.
(a) If the person watches the program on Monday, what is the probability that the person
watches the program on Wednesday?
(b) If 20% of the population watches the program on Thursday, what percentage can be
expected to watch on Friday?
31)
32)
A platypus net is set out on four successive days where a capture–recapture study is
located. Only 1 platypus can be netted per day, and the probability that a platypus will
enter the trap and be netted is 1
15. If X is the number of platypuses captured over the four
days, find the distribution of X.
32)
33)
If a fair coin is tossed six times, determine the probability that at least two heads show.
33)
34)
From a standard deck of 52 playing cards, two cards are randomly drawn in succession
without replacement. If X is the number of red cards that are drawn and f is the
distribution of X, find f(0), f(1), and f(2).
34)
35)
An urn contains two red and eight green marbles. Four marbles are randomly drawn in
succession with replacement. Determine the probability that exactly two marbles are red.
35)
36)
A fair die is rolled four times. What is the probability that exactly three 2′s appear?
36)
37)
At the time of purchase of its product, a company offers a contract to its customers that
guarantees the replacement of the product if it malfunctions within a one–year period. The
cost of the contract is $10. The probability that a unit of the product malfunctions within
one year of purchase is 0.06. If the cost to the company for replacement of a unit is $160,
determine the company’s expected gain or loss per contract.
37)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine the given probability if X is a binomial random variable, n is the number of trials, and p is the probability of
success on any trial.
38)
P(X = 2); n = 10, p =1
3
38)
A)
0.2156
B)
0.1951
C)
0.1929
D)
0.0028
B
Find the standard deviation, , for the binomial distribution which has the stated values of n and p. Round your answer
to the nearest hundredth.
39)
n =46; p =3
5
39)
A)
=3.32
B)
=0.91
C)
=6.59
D)
=7.44
A
Determine the given probability if X is a binomial random variable, n is the number of trials, and p is the probability of
success on any trial.
40)
P(X = 2); n = 5, p = 0.70
40)
A)
0.700
B)
0.132
C)
0.198
D)
0.464
B
41)
P(X =6); n =13, p = 0.5
41)
A)
0.2095
B)
0.3142
C)
0.0156
D)
0.2723
Find the mean, µ, for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth.
42)
n =46; p =3
5
42)
A)
µ=27.9
B)
µ=27.6
C)
µ=28.3
D)
µ=27.1
B
Find the standard deviation, , for the binomial distribution which has the stated values of n and p. Round your answer
to the nearest hundredth.
43)
n =42; p = 0.2
43)
A)
=5.86
B)
=6.71
C)
=2.59
D)
=0.18
C
Find the mean, µ, for the binomial distribution which has the stated values of n and p. Round answer to the nearest tenth.
44)
n =20; p = 0.2
44)
A)
µ=3.5
B)
µ=4.0
C)
µ=4.7
D)
µ=4.3
B
Determine the given probability if X is a binomial random variable, n is the number of trials, and p is the probability of
success on any trial.
45)
P(X = 3); n = 4, p =1
6
45)
A)
0.0039
B)
0.0116
C)
0.0231
D)
0.0154
D
A
Answer Key
Testname: C9
11
Answer Key
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