Long shot, 5 chips aiming for 40

Question

You start with just 5 chips and want to reach 40 chips. Each round is fair: win 1 chip or lose 1 chip with probability $1/2$ each. You stop at 40 (win) or 0 (broke).

What is the probability you reach 40 chips before going broke?

Accepted Answer

Let hi be the probability of reaching N = 40 before 0, with h0 = 0 and h{40} = 1. A fair round gives hi = 	frac12 h{i+1} + 	frac12 h{i-1}, so the win probability is linear in the starting point, hi = i/N. Here h5 = \frac{5}{40} = \frac18. Even though the game is perfectly fair round by round, starting far from the goal makes success a long shot: only a 1/8 chance. Takeaway. Fair does not mean even; your chance is start over target, 5/40 = 1/8, and a distant goal makes ruin the heavy favorite.

Long shot, 5 chips aiming for 40

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