2025 TJC P2 Q7

A factory produces porcelain bowls. It is known that, on average, \(8\%\) of the bowls are faulty. The bowls are packed in boxes of \(20\). A box is considered imperfect if it contains more than \(2\) faulty bowls. Assume that the number of faulty bowls in a box follows a binomial distribution.

1. Find the probability that a randomly chosen box is imperfect.[2]

These boxes are packed into cartons of \(12\) boxes each.

2. Find the probability that, out of \(3\) randomly chosen cartons, there are \(2\) cartons that contains fewer than \(15\) faulty bowls each.[3]

Let \(Y\) be the number of imperfect boxes in a randomly chosen carton.

3. State the values of \(\mathrm{E}(Y)\) and \(\mathrm{Var}(Y)\).[2]
4. Hence using a suitable approximation, find the probability that the total number of imperfect boxes in a random sample of \(35\) cartons is at most \(85\).[2]