134
Chapter 2 Discrete Random Variables and Probability Distributions
1 3
5 7
9 11
13 15
17 19
21 n
2 4
6 8
10 12
VarX
Figure 2 . 1 6
Variance in the candy jars problem.
b. Suppose that attempts are made until 3 successful launchings have occurred.
What is the probability that exactly six attempts will be necessary?
4. A box of manufactured parts contains 4 good and 3 defective parts. They are drawn
out one at a time, without replacement. Let X denote the number of the drawing on which the first defective part occurs.
a. Find the probability distribution for X. b. Find E. X.
5. The probability that a player wins a game at a single trial is
1 3
. Assume the trials follow the binomial assumptions. If the player plays until he wins, find the
probability that the number of trials is divisible by 4.
6. The probability a new driver will pass a driving test is 0.8. a. One student takes the test until she passes it. What is the probability it will take
at least 2 attempts to pass the test?
b. Now suppose 3 students take the driving test until each has passed it. What
is the probability that exactly 1 of the 3 will take at least two attempts before passing the test? Assume independence.
7. To become an actuary, one must pass a series of 9 examinations. Suppose that
60 of those taking each examination pass it and that passing the examinations are independent of one another. What is the probability a person passes the ninth
examination, and thus has passed all the examinations, on the 15th attempt?
2.9 Geometric and Negative Binomial Distributions
135 8. A quality control inspector on a production line samples items until a defective
item is found.
a. If the probability an item is defective is 0.08, what is the probability that at
least 10 items must be inspected?
b. Suppose now that the 16th item inspected is the first defective item found. If
p is the probability an item is defective, what is the value of p that makes the probability that the 16th item inspected is the first defective item found most
likely?
9. A fair coin is tossed. What is the probability the fourth head is preceded by, at
most, 2 tails?
10. A TV interviewer must conduct 5 interviews. Suppose that the probability a person
agrees to be interviewed is
2 3
.
a. What is the probability that the interviewer will ask 9 people in all to be
