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Chapter 2 Discrete Random Variables and Probability Distributions
1 defective, a second sample of 7 is drawn; the lot is accepted if the second sample contains no more than 1 defective and, otherwise, the lot is rejected. Suppose that
at any stage an unacceptable item is replaced by a good item.
a. What is the probability that the lot is accepted? b. What is the average outgoing quality if all the defective items in unacceptable
lots are replaced by good items ?
c. Show the operating characteristic curve for the sampling plan. 12. A day’s production of 200 compact discs is inspected as follows. If an initial sample
of 15 shows, at most, 2 defective discs, the lot is accepted and is subject to no more sampling. However, if the first sample shows 3 or more defective discs, then
a second sample of 20 discs is chosen and the lot is accepted if the total number of defectives in the two samples is no more than 4.
a. Find the probability that the lot is accepted if, in fact, it contains 10 defective
discs.
b. Find the average outgoing quality. c. Plot the operating characteristic curve.
13. A random sample of 100 items is chosen from a lot of 4500 items that is 2
defective. If the sample contains no more than 4 defective items, the lot is accepted; otherwise, the remainder of the lot is inspected and defective items are replaced
by good items.
a. What is the average number of items inspected? b. Graph the average number of items inspected as a function of the percentage
defective in the lot.
2.12
c
The Hy pergeometric Random Variable; Further Examples
Ex ample 2.12.1 A Lotte ry
Lottery games have become popular in many states. In Indiana, the game is played as follows: A player chooses five different numbers from the
integers 1,2,
: : :
,45. Another integer is then chosen from the same set; this choice, called a powerball, may match one of the first five integers chosen.
Lottery officials then choose five integers and the powerball.
2.12 The Hypergeometric Random Variable; Further Examples
157
The number of integers the player correctly chooses from among the first five is a hypergeometric random variable which we call X. Then
P. X = x = 5
x ·
40 5 − x
45 5
; x = 0; 1; : : : ; 5:
Let Y denote the number of correct powerball choices made. Then
P.Y = y = 1
y ·
44 1 − y
45 1
; y = 0; 1:
Since the choices are independent, it follows that
P. X = x and Y = y = 5
x ·
40 5 − x
45 5
· 1
y ·
44 1 − y
45 1
;
x = 0; 1; : : : ; 5; y = 0; 1: Here is a table of values of X and Y giving the probabilities with which
the possible values occur, along with the payoffs to the player. The jackpot varies from week to week.
X Y
Probability Payoff
X Y
Probability Payoff
5 1
1 54;979;155
Jackpot 2
1 19;760
10;995;831 5
5 4
4;998;105 100,000
2 79;040
999;621 4
1 40
10;995;831 5,000
1 1
91;390 10;995;831
2 4
160 999;621
100 1
366;560 999;621
3 1
520 3;665;277
100 1
73;112 6;108;795
1 3
2080 333;207
5 292;448
555;345
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Chapter 2 Discrete Random Variables and Probability Distributions
