4-6. Exercises

Ex 1. A company (the producer) supplies microprocessors to a manufacturer (the consumer) of electronic equipment. The microprocessors are supplied in batches of 50. The consumer regards a batch as acceptable provided that there are not more than 5 defective microprocessors in the batch. Rather than test all of the microprocessors in the batch, 10 are selected at random and tested.

1. Find the probability that out of a sample of 10, d = 0, 1, 2, 3, 4, 5 are defective when there are actually 5 defective microprocessors in the batch.

2. Suppose that the consumer will accept the batch provided that not more than m defectives are found in the sample of 10.

a) Find the probability that the batch is accepted when there are 5 defectives in the batch. ​

b) Find the probability that the batch is rejected when there are 3 defectives in the batch.

[ Solution ]

1. $P(X=d) = \frac {_{45}C_{(10-d)} * _{5} C _{d}} {_{50} C _{10} }$

2. a) $\Sigma P(X=d) = \Sigma \frac {_{45}C_{(10-d)} * _{5} C _{d}} {_{50} C _{10} }$ , $m ≤ 5,$

b) $d =3,$ $P(reject \space batch \space with \space 3 \space defects) =$ $1- \Sigma P(X=d) = 1- \Sigma \frac {_{45}C_{10-d} * _{5} C _{d}} {_{50} C _{10} }$ , $m ≤ 3$

Ex 2. A company buys batches of n components. Before a batch is accepted, m of the components are selected at random from the batch and tested. The batch is rejected if more than d components in the sample are found to be below standard.

(a) Find the probability that a batch which actually contains six below-standard components is rejected when n = 20, m = 5 and d = 1.

(b) Find the probability that a batch which actually contains nine below-standard components is rejected when n = 30, m = 10 and d = 1.

[ Solution ]