2022 ACJC J2 MYE P1 Q10

A factory produces electrical fuses by an automatic process. The fuses are packed in boxes, each box containing \(24\) fuses, for dispatch to customers. The production process is under statistical control and it is known from experience that on average \(p\%\) of the fuses manufactured are defective. It should be assumed that the number of defective fuses in a box of \(24\) fuses follows a binomial distribution.

1. The probability that a randomly selected box contains at least one defective fuse is \(0.518\). Form an equation in \(p\) and hence solve for \(p\), leaving your answer to \(4\) decimal places.[3]
2. Given that \(p=3\) and the customer buys \(100\) boxes of fuses, find the most likely number of boxes each of which contains less than \(1\) defective fuse.[2]