Question

Problem 1. Suppose we are betting money on the outcome of a game of chance with two outcomes (e.g. roulette). If we guess correctly we get double our bet back and otherwise we lose the money we've bet. Consider the strategy where you initially bet one euro and you keep playing and doubling your bet until the first time you win. At that point you go home, having made a net profit. Let p be the probability of winning a single bet, let R denote the total number of rounds you'll play, let B be the amount you bet in your very last game and let P denote the "net profit" you'll have when you go home. Initially we assume you can keep playing (and can keep doubling your bet) indefinitely. (i) What are EP, Var(P), ER, Var(R) and EB, Var(B)? Of course, in practice you do not have unlimited funds. Let M denote the total amount of money you have available. Once your current bet is strictly larger than M/2 then if you loose, you won't be able to keep betting and you'll go home with negative net earnings. For convenience we assume that M is a power of two, i.e. M 2k for some kEN ii) Now what are EP and ER?

Answer 1

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