2024 YIJC H1 Q9

A shop sells plums in bags of \(9\). Plums are taken from a large supply and the probability that a plum is rotten is \(p\), where \(0 < p < 0.2\).

1. It is known that the variance of the number of rotten plums in a bag is \(0.6624\). Find \(p\).[2]

It is now given that \(p = 0.08\).

2. Find the probability that a randomly selected bag of plums has at least \(3\) rotten plums.[2]
3. Five bags of plums are randomly selected. Find the probability that the fifth selected bag is the second bag with at least \(3\) rotten plums.[2]
4. The shop sells \(30\) bags of plums to a customer. Find the probability that less than \(90\)% of the bags have fewer than \(3\) rotten plums.[3]