A player has only one money unit and wishes to increase his fortune to five units. To do so, he plays independent repetitions of a game that, in case of a win, yields double the sum he betted. In case of a loss, he loses his bet. On each play, he bets an amount that, if he wins, enables him either to exactly reach his objective or to get as close as possible to it (for example, if he has three units, then he bets only one). We suppose that the game ends when the player either has achieved his target or has been ruined and that he has a probability equal to 1/2 of winning an arbitrary play.
(a) What is the probability that he reaches his target?
(b) What is the average number of repetitions needed for the game to end?