Dear future self,
I’ve just lost (again) about half an hour of my life trying to find a vaguely remembered formula that generalizes the law of total probability to the case of conditional probabilities. Here it is. You’re welcome.
The law of total probability says that if you can decompose the set of possible events into disjoint subsets (say $B$ and $\overline{B}$), then (with obvious generalization to more than two subsets):
$$\Pr(A) = \Pr(A \mid B) \Pr(B) + \Pr(A \mid \overline{B}) \Pr(\overline{B})$$
But what if you’re dealing with $\Pr(A \mid C)$ instead of just $\Pr(A)$? What’s the formula for the law of total probability in that case? What you’re searching for can be found by googling for “total law probability conditional”:
$$\Pr(A \mid C) = \Pr(A \mid B, C) \Pr(B \mid C) + \Pr(A \mid \overline{B}, C) \Pr(\overline{B} \mid C) $$
There’s a great derivation here: https://math.stackexchange.com/questions/2377816/applying-law-of-total-probability-to-conditional-probability.