2020 DHS P2 Q8

An air force consists of a large number of fighter planes. Each plane carries and releases \(8\) bombs at one time to destroy a designated enemy target. Based on past records, the proportion of bombs hitting an enemy target is \(p\).

1. State, in the context of the question, two assumptions needed for the number of bombs from a plane to hit an enemy target to be well modelled by a binomial distribution.[2]

Assume now that, \(X\), the number of bombs from a plane hitting an enemy target has a binomial distribution.

2. If \(\mathrm{E}(X)+\mathrm{Var}\,(X)=6.5\), show that \(p=0.567\) corrected to \(3\) decimal places.[2]
3. Out of \(30\) randomly chosen planes that were dispatched, estimate the probability that the average number of bombs per plane hitting their targets is less than \(5\).[2]

A mission is considered a success if an enemy bridge is destroyed.

4. \(4\) bombs must hit a bridge before it is destroyed. Find the probability of a successful mission for a plane.[2]
5. \(20\) planes are randomly dispatched to destroy \(1\) bridge each. Find the probability that at least \(13\) but fewer than \(19\) planes’ missions are successful.[2]