I assume that you have to bet your bankroll one dollar at a time, so that it takes m gambles to dispose of a $m bankroll, and that each chip you receive is worth one dollar.
ObPuzzles:Betting a dollar gets you an expected return of one dollar. Therefore, given your bankroll at time t, your expected bankroll at time t+1 is the same. In the long run, you neither gain nor lose.
1. (easy) In the long run, do you expect to gain or lose?
2. (easy) There's nothing special about the number 100; it can WLOG tend to infinity without affecting much.That's true. In that case, the return from a $1 bet is a Poisson-distributed random variable with mean 1, so the return from m $1 bets is the sum of m of these variables, which is Poisson with mean m.
3. (hard) Approximately what are the chances that you are bankrupt after t time units? [I assume that this is in the 100 -> infinity limit.]From the above, if you start with $m, the chance of having $n at the next time unit is e^(-m) m^n/n! . From this, it's an easy induction that if you start with $m, your chance of going bankrupt in t time steps is e^(-m alpha[t]), where
As t becomes large we have
alpha[t] = 2/t + O(log t/t^2),
so your chance of going bankrupt after t time units is
1 - 2/t + O(log t/t^2),
for big t.
David Moews ( email@example.com )
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