2025 EJC P2 Q6

An online game consists of three stages. All players start with Stage \(1\), and no matter whether they win or lose, they proceed to Stage \(2\). Players are allowed to proceed to Stage \(3\) only if they have won at least one of the first two stages.

Ming’s probability of winning Stage \(1\) is \(p\).

If Ming wins a stage, his probability of winning the next stage is halved. If he loses a stage, his probability of winning remains unchanged in the next stage.

1. Draw a probability tree diagram to represent all possible outcomes for Ming over the three stages.[3]
2. Find the range of possible values of \(p\) if the probability that Ming wins all three stages is more than \(0.1\).[2]
3. It is now known that \(p=0.9\). Given that Ming proceeds to Stage \(3\), find the probability that he wins exactly two stages out of the three.[4]