The Hypergeometric Random Variable; Acceptance Sampling Let X denote the total actual value of the gems selected. Find the probability Find E. X. 3. a. A box contains 3 red and 5 blue marbles. The marbles are drawn out one

2.10 The Hypergeometric Random Variable; Acceptance Sampling

143 Exercises 2.10

1. A lot of 50 fuses is known to contain 7 defectives. A random sample of size 10 is

drawn without replacement. What is the probability the sample contains at least 1 defective fuse?

2. A collection of 30 gems, all of which are identical in appearance and are supposed to

be genuine diamonds, actually contains 8 worthless stones. The genuine diamonds are valued at 1200 each. Two gems are selected.

a. Let X denote the total actual value of the gems selected. Find the probability

distribution function for X.

b. Find E. X. 3. a. A box contains 3 red and 5 blue marbles. The marbles are drawn out one

at a time and without replacement, until all of the red marbles have been selected. Let X denote the number of drawings necessary. Find the probability distribution function for X. b. Find the mean and variance for X. 4. a. A box contains 3 red and 5 blue marbles. The marbles are drawn out one at a time and without replacement, until all the marbles left in the box are of the same color. Let X denote the number of drawings necessary. Find the probability distribution function for X. b. Find the mean and variance for X. 5. A lot of 400 automobile tires contains 10 with blemishes which cannot be sold at full price. A sampling inspection plan chooses 5 tires at random and accepts the lot only if the sample contains no tires with blemishes. a. Find the probability that the lot is accepted. b. Suppose any tires with blemishes in the sample are replaced by good tires if the lot is rejected. Find the average outgoing quality of the lot.

6. A sample of size 4 is chosen from a lot of 25 items of which D are defective. Draw

the curve showing the probability that the lot is accepted as a function of D if the lot is accepted only when the sample contains no defective items.

7. A lot of 250 items that contains 15 defective items is subject to an acceptance

sampling plan that calls for a sample of size 6 to be drawn. The lot is accepted if the sample contains, at most, one defective item. a. Find the probability that the lot is accepted. b. Suppose any defective items in the sample are replaced by good items. Find the average outgoing quality. 144

Chapter 2 Discrete Random Variables and Probability Distributions