2018 PJC BT2 P2 Q8

Tomatoes are sold in boxes of \(16\). On average, \(15\%\) of them are damaged.

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

Assume now that the number of damaged tomatoes in a box has a binomial distribution.

2. Find the most likely number of damaged tomatoes in a randomly chosen box.[2]
3. Find the probability that a box contains less than \(6\) damaged tomatoes.[1]
4. A box contains less than \(6\) damaged tomatoes. Find the probability that it contains at least \(2\) damaged tomatoes.[3]
5. Two boxes of tomatoes are randomly selected. Find the probability that one box has at least \(2\) damaged tomatoes and the other box has none.[2]
6. A market trader buys \(80\) boxes of tomatoes. Find the probability that there are more than \(60\) boxes with at least \(2\) damaged tomatoes, giving your answers to \(4\) decimal places.[3]