2024 SAJC P2 Q5

A dart player, Mike, in Smart Andrew’s College Darts Team is having a practice session of dart throwing in the hall. The probability that Mike hits his target on the first throw is \(0.7\).

For each throw after his first throw, the probability that Mike hits the target, given that he hit the target on his preceding throw, is \(0.77\).

For each throw after his first throw, the probability that Mike does not hit the target, given that he did not hit the target on his preceding throw, is \(0.33\).

1. Draw a fully labelled tree diagram to represent Mike’s first two throws.[2]
2. Find the probability that Mike hits his target on the second throw of the dart.[2]

It is now known that Mike threw the dart three times.

3. Given that Mike did not hit his target on the second throw, find the probability that he hit the target twice.[3]