Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pairs (1, 2) and (2, 1) represent the selection of boards 1 and 2 for inspection. (a) List the ten different possible outcomes. (Enter your answers as a comma-separated list of ordered pairs.) (b) Suppose that boards 1 and 4 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X.

Explanation:

There are total 5 batches and two boards are selected from each batch for inspection.

Let the boards are numbered from 1 to 5.

If the selected boards are 1 and 2, then it is represented in pair as (1, 2).

If the selected boards are 1 and 3, then it is represented in pair as (1, 3).

Similarly, other pairs can be obtained.

a) Let X be the number of defective boards observed among the two inspected.

If the boards 1 and 2 are the only defective boards in a lot of five, then

(1,2),x=2; (1,3),x=1; (1,4),x=1; (1,5),x=1;

(2,3),x=1; (2,4),x=1; (2,5),x=1;

(3,4),x=0;\ (3,5),x=0;\ (4,5),x=0.(3,4),x=0; (3,5),x=0; (4,5),x=0.

P(X=0)=  3/10 =0.3

P(X=1)={6 \over 10}=0.6P(X=1)=  6/10 =0.6

P(X=2)={1 \over 10}=0.1P(X=2)=  1/10 =0.1

b)

x                  0               1                    2

p(x)            0.3           0.6                0.1

μ  X  =0⋅0.3+1⋅0.6+2⋅0.1=0.8

σ  X ² =(0−0.8)²  ⋅0.3+(1−0.8)²  ⋅0.6+(2−0.8)² ⋅0.1=0.36

μₓ=0.8, σX  =0.6

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