Consider the doubling strategy in the case of a game of roulette. Assume that a gambler always bets on red. Then at each play, the probability of success is p = 9/19 (there are 18 red, 18 black, and 2 green cells).

(a) Is the profit Wt a martingale?

(b) What will happen if after each failure, the gambler bets the amount c times as large as in the previous play? Consider the cases 1 < c < 2 and c > 2 separately.

(c) Let us consider the situation. Suppose that once the gambler quits (either winning one or losing being not able to apply the doubling strategy), he starts over with a stake of one and doubling the bet after each loss; that is, he repeats the whole game (that is, the whole sequence of plays up to quitting). Using the LLN, analyze the behavior of the total profit for a long sequence of such (independent) games. (To make your result consistent with the answer to the next problem, recalculate the probability of losing with a larger accuracy.)

(d) Solve Problem 19c for the general case where the probability of winning in a separate try equals a (given) p > 0, and the gambler is not able to apply the doubling strategy if the maximal bet is 2^m.