Quant Interview Question #quant

Two players play a game. Player one is first given two independent random numbers, each uniformly distributed between zero and one. Player two cannot see these numbers. Then, player two is given a random number, also uniformly distributed between zero and one. After seeing his own number, he may either keep it or re-roll once, in which case he must keep the second number. If player two's final number lies between player one's two numbers, player two wins. Otherwise, player one wins. Assuming both players play optimally, what is the probability that player two wins? Post your answers in comments.