7. To be a winner in the following game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on (0, 1). If U > 1, then you are successful in round 1; if U > 2, then you are successful in round 2; and if U > 3, then you are successful in round 3.

(a) Find the probability that you are successful in round 1.

(b) Find the conditional probability that you are successful in round 2 given that you were successful in round 1.

(c) Find the conditional probability that you are successful in round 3 given that you were successful in rounds 1 and 2.

(d) Find the probability that you are a winner.