Climbing a ladder of three rungs

Question

A player passes through stages P, Q, R. From P: advance to Q with probability 0.5, otherwise Lose (probability 0.5). From Q: advance to R with probability 0.5, otherwise slip back to P (probability 0.5). From R: Win with probability 0.5, otherwise slip back to Q (probability 0.5). Win and Lose end the game. Starting from P, what is the probability the player Wins before Losing?

Accepted Answer

Let f(P), f(Q), f(R) be the win-before-lose probabilities, with f(	ext{Win}) = 1 and f(	ext{Lose}) = 0. First-step analysis gives: f(P) = 0.5\,f(Q) + 0.5 \cdot 0 = 0.5\,f(Q), f(Q) = 0.5\,f(R) + 0.5\,f(P), f(R) = 0.5 \cdot 1 + 0.5\,f(Q) = 0.5 + 0.5\,f(Q). Substitute f(R) into the f(Q) equation: f(Q) = 0.5\,(0.5 + 0.5\,f(Q)) + 0.5\,f(P) = 0.25 + 0.25\,f(Q) + 0.5\,f(P). Since f(P) = 0.5\,f(Q), the last term is 0.5 	imes 0.5\,f(Q) = 0.25\,f(Q): f(Q) = 0.25 + 0.25\,f(Q) + 0.25\,f(Q) = 0.25 + 0.5\,f(Q) \implies 0.5\,f(Q) = 0.25 \implies f(Q) = 0.5. Then f(P) = 0.5 	imes 0.5 = 	frac14. So the p…

Climbing a ladder of three rungs

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