2020 YIJC P2 Q8

A manufacturing company produces surgical masks. The surgical masks are randomly packed into boxes of \(50\). On average, \(15\%\) of the surgical masks are defective. The number of surgical masks that are defective in a box of \(50\) pieces is denoted by \(X\).

1. State, in context, two assumptions needed for the number of defective surgical masks in a box to be well modelled by a binomial distribution.[2]

Assume now that the number of defective surgical masks in a box has a binomial distribution.

In a randomly chosen box of \(50\) surgical masks, find the probability that there are

2. more than \(8\) defective surgical masks,[2]
3. at most \(12\) defective surgical masks given that there are more than \(8\) defective surgical masks.[3]

For shipping purposes, the boxes are packed into cartons, with each carton containing \(24\) boxes.

4. Find the probability that, in a randomly chosen carton, there are at least \(10\) boxes containing at most \(8\) defective surgical masks.[2]

The company also manufactures reusable masks which are packed into packets of \(10\). The probability that a reusable mask is defective is \(p\). It is known that the modal number of defective reusable masks in a packet is \(1\).

5. Use this information to find exactly the range of values that \(p\) can take.[3]