2024 ASRJC H1 Q7

In a computer game played by a single player, the objective is to find a path through a maze shown on the screen within a fixed time. On the first occasion, the computer presents a simple maze, and the probability that the player succeeding within the allowed time is \(\frac{4}{5}\). On the subsequent occasions, the complexity of the maze depends on the result of the previous result. If the player succeeded in the previous occasion, the next maze is more difficult, and the probability of success is halved. If the player failed on the previous occasion, a simple maze is shown again with the probability of success as \(\frac{4}{5}\).

The player plays three games.

1. Draw a fully labelled tree diagram to represent all the possible outcomes for the three games.[2]
2. Show that the probability that the player succeeds in exactly two of the games is \(\frac{88}{125}\).[2]
3. Find the probability that the player has success in first game given that the player does not have exactly two successes.[3]