2022 ASRJC BT2 P2 Q6

On average, it is known that \(1\) in \(12\) peaches produced in a farm are classified as very sweet. A farmer randomly selected \(40\) peaches from the farm.

1. State, in this context, two necessary assumptions to model the random variable representing the number of very sweet peaches by a binomial distribution.[2]

The farmer randomly selects \(5000\) peaches and packs them into boxes of \(40\) peaches each for shipment.

2. Obtain an approximate distribution for the mean number of very sweet peaches in a box for this shipment. Hence find the probability that this mean number is at most \(3\).[3]
3. What is the least number of boxes that the farmer must ship so that the probability of obtaining not less than \(200\) very sweet peaches is more than \(0.99\) for this new shipment?[4]