2022 NJC P2 Q7

On average, \(7.8\%\) of markers produced by a factory are faulty. Each day, the factory’s quality manager picks a random sample of n markers and inspects the number of faulty markers found in the sample. The number of faulty markers found in the sample is the random variable \(X\) and whether the markers are faulty are independent of one another.

1. Explain what is meant by a random sample in this context.[1]
2. Given that the probability that more than \(\left( n-4 \right)\) markers in the random sample are found to be non-faulty is less than \(0.3\), find the least possible value of \(n\).[2]

Assume for the remainder of this question that \(n=10\).

The samples in \(50\) randomly chosen days are collected and the number of faulty markers in each sample is recorded.

3. Find the probability that exactly \(35\) of the samples each contains at least one faulty marker, given that none of the \(50\) samples contain more than \(2\) faulty markers each.[4]