How many bets until the gambler stops?

Question

A gambler starts with \40 and makes repeated fair \1 bets (win or lose a dollar with probability 	frac12 each). She stops when she goes broke at \0 or reaches her target of \100. What is the expected number of bets she makes?

Accepted Answer

Shift coordinates so the walk starts at 0: the lower barrier (broke) is a = 40 below her, and the upper barrier (target) is b = 60 above. Her wealth is then a symmetric \pm 1 walk absorbed at -40 or +60. For a symmetric walk, Mt = St^2 - t is a martingale (the drift of St^2 is exactly +1 per step, which -t cancels). Optional stopping at the exit time T gives E[ST^2] - E[T] = 0 \implies E[T] = E[ST^2]. The exit-probability result says she reaches +60 with probability 	frac{a}{a+b} = 	frac{40}{100} and -40 with probability 	frac{60}{100}, so E[T] = \frac{40}{100}(60)^2 + \frac{60}{100}(40)^2…

How many bets until the gambler stops?

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