2025 TMJC P2 Q8

A manufacturer produces mystery boxes, each containing either a regular or a seasonal toy. On average, \(p\)% of the mystery boxes contain a seasonal toy.

Amber orders \(n\) mystery boxes from the manufacturer. The number of seasonal toys that Amber gets is the random variable \(S\).

1. State, in the context of the question, two assumptions needed for \(S\) to be well modelled by a binomial distribution.[2]

You are now given that \(S\) can be modelled by a binomial distribution.

2. Given that \(\mathrm{P}(S=2)=\mathrm{P}(S=3)\) and \(\mathrm{E}(S)=2.96\), find the value of \(p\).[4]

Assume now that \(p=5\).

The manufacturer now packs the mystery boxes into cartons of \(12\) each for sale. Each carton is checked for quality control. If there is at most \(1\) mystery box containing a seasonal toy, the carton is accepted. Otherwise, the carton is rejected.

3. Given that a randomly chosen carton is rejected, find the probability that no more than \(30\)% of the boxes in the carton each contains a seasonal toy.[3]