2017 NJC P2 Q10

Factory \(A\) manufactures a large batch of light bulbs. It is known that on average, \(1\) out of \(200\) light bulbs manufactured by Factory \(A\), is defective. A random sample of \(180\) light bulbs is inspected. The batch is accepted if the sample contains less than \(r\) defective light bulbs.

1. Explain why the context above may not be well-modelled by a binomial distribution.[1]

Assume now that the context above is well-modelled by a binomial distribution.

2. Determine the value of \(r_{{}}^{{}}\) such that the probability of accepting the batch is \(0.998\).[1]

In Factory \(B\), a random sample of \(30\) light bulbs is taken from a large batch. If the sample contains no defective light bulbs, the batch is accepted. The batch is rejected if the sample contains more than two defective light bulbs. If the sample contains one or two defective light bulbs, a second random sample of \(30\) light bulbs is chosen and the batch is accepted only if this second sample contains no defectives. It is known that Factory \(B\) produces \((100p)%\) defective light bulbs.

3. Find the probability that the batch is accepted. Leave your answer in terms of \(p\).[3]

Forty random samples of \(30\) light bulbs are taken from each of the two factories \(A\) and \(B\).

4. Given that \(p=0.007\) and there is exactly one defective bulb, find the probability that it is from Factory \(B\).[4]