In a computer game played by a single player, the player has to find, within a fixed time, the path through a maze shown on the computer screen. On the first occasion that a particular player plays the game, the computer shows a simple maze, and the probability that the player succeeds in finding the path in the time allowed is. On subsequent occasions, the maze shown depends on the result of the previous game. If the player succeeded on the previous occasion, the next maze is harder, and the probability that the player succeeds is one half of the probability of success on the previous occasion. If the player failed on the previous occasion, a simple maze is shown and the probability of the player succeeding is again.

The player plays three games.

(a) Show that the probability that the player succeeds in all three games is

(b) Find the probability that the player succeeds in exactly one of the games.

(c) Find the probability that the player does not have two consecutive successes.

(d) Find the conditional probability that the player has two consecutive successes given that the player has exactly two successes. (C)