How long does a losing streak at the casino last?

Question

You start with i chips and bet 1 chip per round. Each round you win a chip with probability p and lose one with probability q = 1 - p, with p 
eq q. You stop when you reach N chips (goal) or 0 chips (broke). What is the expected number of rounds you play? Then evaluate it for p = 0.4, N = 10, i = 5.

Accepted Answer

Let Di be the expected number of rounds starting from i chips, with boundary conditions D0 = DN = 0. One round happens, then you are at i \pm 1: Di = 1 + p\,D{i+1} + q\,D{i-1}. This is a linear recurrence with a constant forcing term. Its solution (the standard biased-ruin duration formula) is Di = \frac{i}{q - p} - \frac{N}{q - p}\cdot\frac{1 - (q/p)^{i}}{1 - (q/p)^{N}}. You can verify the boundaries: at i = 0 both terms vanish, and at i = N the two terms cancel, so D0 = DN = 0. The first term is the "pure drift" estimate (distance to travel divided by drift per step); the second corrects fo…

How long does a losing streak at the casino last?

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