2022 EJC P2 Q6

A company manufactures screen protectors for handphones. On average, \(17\%\) of the screen protectors are cracked. The screen protectors are packaged into boxes of \(20\). It should be assumed that the number of cracked screen protectors in a box of \(20\) screen protectors follows a binomial distribution.

1. Show that the probability of having no more than \(3\) cracked screen protectors in a randomly chosen box is \(0.55041\), correct to \(5\) significant figures.[1]
2. Find the most probable number of cracked screen protectors in a randomly chosen box.[1]

Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are \(8\) boxes of screen protectors. Every carton is assumed to be filled completely.

3. Using a suitable approximation, find the least value of \(n\) such that in a randomly chosen month, there is at least a \(99\%\) chance that the average number of boxes per carton having no more than \(3\) cracked screen protectors is at most \(5\).[4]

To determine whether cartons are ready for local or overseas sales, cartons are randomly selected for inspection. In a carton,

• if there are at least \(4\) boxes that contain no more than \(3\) cracked screen protectors each, the carton is ready for local sales;

• if there are at least \(6\) boxes that contain no more than \(3\) cracked screen protectors each, the carton is ready for overseas sales.

4. A carton that is not ready for overseas sales is randomly selected. Find the probability that the carton is ready for local sales.[3]