My thanks to anyone who can help with the following problem. It's an analog to a behavioral assessment (of basic discrimination ability) with a "pass" criterion equivalent to the "win" criterion described below. We have used the assessment for many years, but to my knowledge no one has articulated the precise probability of passing by chance. # The Game # Toss a fair coin repeatedly, recording the result on each toss. The game can end in two ways: 1. You land eight heads consecutively, thereby winning. 2. You land eight tails cumulatively, thereby losing. The shortest possible game is 8 tosses: TTTT TTTT or HHHH HHHH. The longest possible game is 64 tosses: HHHH HHHT HHHH HHHT HHHH HHHT HHHH HHHT HHHH HHHT HHHH HHHT HHHH HHHT HHHH HHH(T/H) What the probability (relative frequency) of winning?