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Something to think about: each flip of the coin where you see heads changes the probability you've got the fake coin. If you flipped the coin 100 times and got heads every time, I think you'd be about 100% certain that the next time it will be heads.
I thought of it the same way Devac thought about it but given the hint it comes down to something like this. Edits for formatting Turn Fake ------- Real Probability 1 1/2 ---- 1 / (.5 +1) 1 1/4 ------- 1/(.25+1 ) 1 1/8 --------- 1/(.125+1 ) 1 1/16 -------- 1/(.0625+1 ) -94% 1 1/2^N ---------- 1/(.5^N+1 )