The first part of the question is to find the probability that the two coins take the same number of flips to land on heads, which I found to be $1/3$.The next part, which I'm stuck on, it to find the conditional probability mass function of $X$ given that both coins land on heads on the same number of flips. Any help?
2026-03-30 15:31:19.1774884679
Flipping two coins; $X$ is how many times first coin is flipped until heads, $Y$ is how many times second coin is flipped until heads
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The solution alluded to by Did in a comment:
$$ \textsf P(X=x\mid X=Y)=\frac{\textsf P(X=x\cap X=Y)}{\textsf P(X=Y)}=\frac{\left(\frac14\right)^x}{\frac13}=\frac3{4^x}\;. $$