Fair coin climb from 30 to 50

Question

You hold 30 points. Each round you bet 1 point on a fair coin, winning or losing 1 point with probability $1/2$ each. You stop at 50 points (win) or 0 points (broke).

What is the probability you reach 50 points before going broke?

Accepted Answer

Let hi be the probability of hitting N = 50 before 0, with h0 = 0 and h{50} = 1. Conditioning on one fair round, hi = 	frac12 h{i+1} + 	frac12 h{i-1}, which forces equal gaps and a linear solution hi = i/N. So h{30} = \frac{30}{50} = \frac35. Starting closer to the top than the bottom, you are the favorite: a 60\% chance of success. Takeaway. For a fair game the win chance is start over target, 30/50 = 3/5; distance to each wall is all that matters.

Fair coin climb from 30 to 50

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