Is it 17/18? EDIT: here's my thought process-- Breakdown of real coin flips: H H H H H T H T H H T T T H H T H T T T H T T T Therefore H H H is one of the eight possibilities for a 3-flip scenario. Breakdown of fake coin flips: H H H H H H H H H H H H H H H H H H H H H H H H Therefore H H H is 8 of the 8 possibilities for a 3-flip scenario. Really it's the only possibility, but I expanded it since we have an equal shot at originally picking each coin before flipping. This means that, of the 3-flip scenarios, 9 of the 16 results involve H H H. We'll limit our probability space to those results. 1 belongs to the real coin, so we'll have a 1/9 shot at having picked that one. We'll then have an 8/9 shot at having picked the fake coin. All the fake coin selections will result in H H H H, so we'll leave that 8/9 as is. Since the real coin has a 1/2 shot at producing H H H H at this point, we'll cut the 1/9 in half. Now we add both fractions together. TLDR: given H H H, we have a 1/9 chance of having picked the real coin and an 8/9 chance of having picked the fake coin. Therefore we have a 1/18 chance of having picked the real coin and then flipping tails, and all the other situations will result in heads.
Just as a pedagogical exercise... you could say that after 4 flips the following are possible: Real coin HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT Fake coin HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH Since we know the first three flips are Heads, we eliminate all cases that don't begin with HHH. Tha leaves us: Real coin: HHHH HHHT Fake coin: HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH HHHH There are 18 possibilities. 17 of them have Heads on the 4th flip. P(H on 4th) = 17/18. Same as what you did, but a little more direct.