Reaching the finish square by coin flips

Question

A tiny board has squares 0, 1, 2, 3, with 0 the start and 3 the finish. Each turn you flip a fair coin: tails moves you 1 square forward, heads moves you 2 squares forward. You finish as soon as you land on or pass square 3 (overshooting is fine). What is the expected number of coin flips to finish, starting from square 0?

Accepted Answer

Let Ek be the expected flips to finish from square k. Squares 3 and beyond are done, so E3 = 0. From square 2: either move gets you to 3 or past it. One flip finishes it: E2 = 1. From square 1: tails 	o square 2; heads 	o square 3 (done). So E1 = 1 + 	frac12 E2 + 	frac12 E3 = 1 + 	frac12(1) + 	frac12(0) = 1.5. From square 0: tails 	o square 1; heads 	o square 2: E0 = 1 + 	frac12 E1 + 	frac12 E2 = 1 + 	frac12(1.5) + 	frac12(1) = 1 + 0.75 + 0.5 = 2.25. So on average it takes 2.25 flips to finish. Intuition: each flip advances you 1.5 squares on average, so covering 3 squares "should…

Reaching the finish square by coin flips

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