Question

A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100% screening. If no defectives are found, the box is shipped.

(a) What is the probability that a box containing 3 defectives will be shipped?

(b) What is the probability that a box containing only 1 defective will be sent back for screening?

Answer 1

Solution

(a) What is the probability that a box containing 3 defectives will be shipped?

Let X the number of defective item. 

If the box containing 3 defective is shipped so. X=0

\( \implies P(X=0)=\frac{C_{22}^3\times C_3^0}{C_{25}^3\:}=\frac{22!\times \:22!}{25!\times \:19!} \)

                             \( =\frac{20\times 21\times \:22}{23\times \:24\times \:25\:}=0.669 \)

Therefore.  \( P(X=0)=0.669 \)

(b) What is the probability that a box containing only 1 defective will be sent back

\( \implies P(X=1)=\frac{C_{24}^2}{C_{25}^3}=\frac{6}{25\times 2}=0.12 \)

Therefore.  \( P(X=1)=0.12 \)

Therefore.

a). \( P(X=0)=0.669 \)

b). \( P(X=1)=0.12 \)