A single player has to complete, in a fixed time, the path through maze shown on the computer screen. On the first occasion the computer shows a simple maze, and probability that the player succeeds in determine the path in the time allowed is 0.75. On subsequent occasions, the difficulty level of maze shown depends upon the result of previous game. If the player succeeded upon the previous occasion, the probability that the player succeeds is one half probability of success on previous occasion. If the player failed on the previous occasion, a simple maze is shown and the probability of the player succeeding is again 0.75. The player plays three games.
(a) Find the probability that the player succeeds in all three games.
(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.