2023 JPJC P2 Q12

A manufacturer produces screen protectors for handphones. On average, \(4\%\) of the screen protectors are cracked. The screen protectors are sold in boxes of \(n\).

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

Assume now that the number of cracked screen protectors in a box has a binomial distribution.

2. It is given that the most likely number of cracked screen protectors in a box is \(2\). Find the range of values of \(n\), without using a calculator.[4]

For the rest of the question, the value of \(n\) is taken to be \(50\).

3. Find the probability that there are at most \(3\) cracked screen protectors in a box.[1]
4. Find the probability that there are more than \(2\) cracked screen protectors in a box, given that there are at most \(6\) cracked screen protectors in a box.[3]
5. \(10\) boxes of screen protectors are randomly chosen. Find the probability that the seventh box is the fourth box with at most \(3\) cracked screen protectors.[2]