2024 CJC H1 Q6

During the preliminary round of an air rifle competition which attracted a large number of participants, each participant is required to execute five shots. The probability that a participant executes a successful shot, i.e. the shot hits the target, during the preliminary round is \(0.38\). Assume that the success of each shot is independent of all other shots.

1. Find the probability that a participant obtains at least three successful shots in the preliminary round.[1]

A participant is eliminated if he obtains less than three successful shots in the preliminary round. Two participants are selected at random.

2. Find the probability that both participants are eliminated.[2]

In the final round, each participant is required to execute \(n\) shots. The probability that a participant executes a successful shot during the final round is \(p\).

3. Given that the mean and variance of the number of successful shots during the final round are \(13.8\) and \(4.278\) respectively, find \(n\) and \(p\).[3]