Math 141 Week-In-Review
7.1-7.6, 8.1-8.4
1. If 65% of a store’s customers are female and 80% of the female customers have charge accounts at the
store, what is the probability that a customer selected at random is a female and has a charge account?
Solution: P(FC) = P(C|F) P(F) = .80(.65) = .52
2. A single card is drawn from a standard 52-card deck. If event E is the drawn card is a spade and event F is
the drawn card is a 8 or 9, are the two events independent?
Solution: P(E) P(F) = (1/4)(8/52) = 1/26
P(EF) = 2/52 = 1/26
Yes, they are independent.
3. If the odds in favor of Jamie attending class on Friday is 7/11, what is the probability that Jamie will skip
class on Friday? Express your answer to the nearest tenth of a percent.
Solution: 7/18  38.9%
4. A fair die is rolled 8 times. What is the probability of getting a 5 on the 8th roll, given that a 5 turned up on
the preceding 7 rolls?
Solution: 1/6
5. A survey indicated that 68% of college students like a certain product. Based on the survey, find the
probability that at least 25 college students in a sample of 30 like the product.
Solution: Binomial Experiment---.0488…
6. Find P(AC) by referring to the following tree diagram:
C
.7
A
.3
.62
D
.45
C
.38
B
.55
D
Solution: P(AC) = P(A)+P(C)-P(AC) = .791
7. Find the probabilities by referring to the following tree diagram:
C
.4
U
.6
.2
CC
C
.5
.2
V
.8
.3
CC
C
.6
W
.4
CC
a.
P(C)
Solution: .08 + .1 +.18 = .36
b.
P(C|U)
Solution: .4
c.
P(VC)
Solution: P(V) + P(C) – P(VC) = .5 + .36 - .1 = .76
d.
P(U|C)
Solution: .08/.36 = 2/9
e.
P(V/ C C )
Solution: .4/(.12+.4+.12) = 1/16
f.
P(VC)
Solution: .1
g.
P(C|W)
Solution: .18
8. The probability of a pencil lead breaking during next test is .085. What are the odds in favor of a pencil lead
breaking during the next test?
Solution: .085/.915 = 17/183
9. A committee of five must be chosen from among 8 girls and 20 boys. What is the probability that at least 3
girls are on the committee? Round your answer to 4 decimal places.
Solution: [C(8,3)C(20,2) + C(8,4)C(20,1) + C(8,5)]/C(28,5) = 8/65  .123
10. Three balls are drawn, at the same time, from a bag containing 4 white and 5 red balls. Let the random
variable X denote the number of red balls drawn.
a. Classify the random variable as a) finite discrete, b) infinite discrete, or c) continuous.
Solution: finite discrete
b. Find the probability distribution of the random variable associated with this experiment.
Solution:
X
P(X)
0
C(5,3)/C(9,3) = 5/42
1
C(4,1)C(5,2)/C(9,3) = 10/21
2
C(4,2)C(5,1)/C(9,3) = 5/14
3
C(4,3)/C(9,3) = 1/21
c. How many white balls can you expect to draw from the box?
Solution: E(X) = 0(5/42) + 1(10/21) + 2(5/14) + 3(1/21) = 4/3
11. Suppose P(EF) = .68, P(E) = .38 and P(F) = .3. Are the events E and F independent? Why or why not?
Solution: P(E) P(F) = .38*.3 = .114
P(EF) = P(EF) – P(E) – P(F) = .68 - .38 - .3 = 0
No, because P(E) P(F)  P(EF)
12. A pair of fair dice is cast. What is the probability that the sum of the numbers falling uppermost is even
given that one of the numbers is a 4?
Solution: 5/11
13. A carton of 25 calculator batteries contains 4 dead ones. A random sample of 5 is selected from the 25
and tested. Let X be the random variable associated with the number of dead batteries found in a sample.
a. Classify the random variable as a) finite discrete, b) infinite discrete, or c) continuous.
Solution: finite discrete
b. Find the probability distribution of X.
Solution:
X
P(X)
0
C(21,5)/C(25,5)  .3830
1
C(21,4)C(4,1)/C(25,5)  .4506
2
C(21,3)C(4,2)/C(25,5)  .1502
3
C(21,2)C(4,3)/C(25,5)  .0158
4
C(21,1)C(4,4)/C(25,5)  .0004
c. Find the expected number of dead batteries found in a sample.
Solution: E(X)  0(.3830) + 1(.4506) + 2(.1502) + 3(.0158) + 4(.0004) = .8
14. What is the expected value (long-run average) of the number of dots facing up for the roll of a single die?
Solution: 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 21/6 = 3.5
15. A spinner device is numbered from 0 to 6, and each of the numbers is as likely to come up as any other. A
player who bets $1 on any given number wins $5 (and gets the bet back) if the pointer comes to rest on
the chosen number; otherwise, the $1 bet is lost. What is the expected value of the game (long-run
average gain or loss per game)?
Solution: E(X) = 5(1/7) – 1(6/7) = -1/7  -14 cents
X
5
-1
P(X)
1/7
6/7
16. Use the data set in the histogram to answer the following questions.
a. What is the median?
Solution: 10
b. What is the variance?
Solution: 5.15
.35
c. What is the standard deviation?
.30
Solution: 2.2694
.25
d. What is the expected value of X?
.20
Solution: .85
.15
e. What is the interquartile range?
.10
Solution: 10-6 = 4
.05
f. What is the mode?
Solution: 10
4 6 8 10 12
An experiment consists of selected a letter at random from the words SAMPLE SPACE.
17. Write the sample space for the experiment.
Solution: {A, C, E, L, M, P, S}
18. How many events are possible for the experiment?
Solution: 2^7 = 128
19. What is the event that a letter chosen in the situation is a vowel?
Solution: {A, E}
20. Let the random variable X denote the number of boys in a six-child family. If the probability of a male birth
is .508,
a. Find the probability of exactly 3 boys in a six-child family.
Solution: Binomial Experiment: .3123
b. Find the probability of at least 4 boys in a six-child family.
Solution: .3589
c. Construct the binomial distribution and draw the histogram associated with the experiment.
Solution:
X
0
1
2
3
4
5
6
P(X)
.0142
.0879
.2268
.3123
.2418
.0999
.0172
.35
.30
.25
.20
.15
.10
.05
0 1 2 3 45 6
d. Compute the mean and standard deviation of the random variable associated with this experiment.
Solution: Mean  3.0480 Standard Deviation  1.2247