14.7 [De Mere’s martingale] We have a fair coin p = 1/2 and a fair game i.e., twice the amount gambled if the toss is won, and lose all amount if the toss is lost.

Suppose we adopt a doubling strategy, that is, we start with x dollars, if we lose, we next bet 2x, and so on. If we lose n bets, the amount bet on the n+1 toss is 2n+1x. We stop playing the moment we win (or equivalently we bet 0 dollars for all subsequent tosses). Let Zn the net profit after the nth toss occurred.

a) Show that Zn is a martingale.

b) Show that the game ends in a finite time almost surely.

c) Calculate the expected time until the gambler stops betting.

d) Calculate the expected net profit at the time when the gambler stops betting.

e) Calculate the expected value of the gambler’s maximum loss during the game. Comment.