2024 TMJC H1 Q9

Apples are delivered to a supermarket in boxes of \(100\). It was found that on average, \(5\)% of the apples delivered were bruised.

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

Assume now that the number of bruised apples in a box has a binomial distribution.

2. Find the probability that a randomly chosen box contains four bruised apples.[1]

The apples are repackaged into bags of \(n\) to be sold at the supermarket.

3. Find the least value of \(n\) such that the probability that a randomly chosen bag contains fewer than two bruised apples is at most \(0.9\).[3]
4. A customer bought \(10\) bags containing \(12\) apples each. Find the mean and variance of the number of bags bought by the customer that contain fewer than two bruised apples.[3]