Home base before jail?

Question

A board-game token starts on square X. From X it reaches Home with probability 0.5 or slides to square Y with probability 0.5. From Y it lands in Jail with probability 0.5 or slides back to X with probability 0.5. Home and Jail end the game. Starting from X, what is the probability the token reaches Home before Jail?

Accepted Answer

Let f(X) and f(Y) be the probabilities of reaching Home before Jail from those squares, with f(	ext{Home}) = 1 and f(	ext{Jail}) = 0. First-step analysis: f(X) = 0.5 \cdot 1 + 0.5\,f(Y) = 0.5 + 0.5\,f(Y), f(Y) = 0.5 \cdot 0 + 0.5\,f(X) = 0.5\,f(X). Substitute the second into the first: f(X) = 0.5 + 0.5\,(0.5\,f(X)) = 0.5 + 0.25\,f(X) \implies 0.75\,f(X) = 0.5 \implies f(X) = \frac{2}{3}. So the token reaches Home before Jail with probability 	frac{2}{3}. Takeaway. Even a tight two-square loop yields to first-step analysis: write each square's win-chance as the average of its next steps and…

Home base before jail?

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