My thanks to anyone who can help with the following problem. It's an\nanalog to a behavioral assessment (of basic discrimination ability)\nwith a "pass" criterion equivalent to the "win" criterion described\nbelow. We have used the assessment for many years, but to my knowledge\nno one has articulated the precise probability of passing by chance.\n\n# The Game #\nToss a fair coin repeatedly, recording the result on each toss. The\ngame can end in two ways:\n1. You land eight heads consecutively, thereby winning.\n2. You land eight tails cumulatively, thereby losing.\n\nThe shortest possible game is 8 tosses: TTTT TTTT or HHHH HHHH.\nThe longest possible game is 64 tosses:\n\nHHHH HHHT HHHH HHHT\nHHHH HHHT HHHH HHHT\nHHHH HHHT HHHH HHHT\nHHHH HHHT HHHH HHH(T/H)\n\nWhat the probability (relative frequency) of winning?