A game character between checkpoints 0 and 8

Question

In a game, a character stands at checkpoint 5 on a line of checkpoints numbered 0 through 8. Each turn a fair coin decides its drift: forward one checkpoint with probability 	frac12, back one with probability 	frac12. The level ends when it reaches checkpoint 8 (victory) or checkpoint 0 (fail). What is the probability the character reaches checkpoint 8 before checkpoint 0?

Accepted Answer

Let pk be the win probability from checkpoint k, with p0 = 0 and p8 = 1. First-step analysis over the fair move gives pk = 	frac12 p{k-1} + 	frac12 p{k+1}, so consecutive differences are equal and pk is linear: pk = k/8. Therefore p5 = \frac{5}{8} = 0.625. Takeaway. Symmetric drift between two absorbing checkpoints yields a straight-line win probability, k/N; here five-eighths of the way across gives odds of 5/8.

A game character between checkpoints 0 and 8

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