(from Hamming26) A game is played as follows: A random number X is chosen uniformly from [0,...

Question:

(from Hamming26) A game is played as follows: A random number X is chosen uniformly from [0, 1]. Then a sequence Y1, Y2, . . . of random numbers is chosen independently and uniformly from [0, 1]. The game ends the first time that Yi > X. You are then paid (i − 1) dollars. What is a fair entrance fee for this game?