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.