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I've taken multiple prob/stats courses, but I saw this simple question, and for some reason, the answer didn't come to me immediately. So I'm looking for the intuition for how to think about this again.

$\text{Give a formula for } P(G|\lnot H) \text{ in terms of } P(G), P(H), \text{ and } P(G \land H) \text{ only.}$ (H and G are boolean random variables).

Here's how I tried to solve it. $P(G | \lnot H) = \frac{P(\lnot H | G)P(G)}{P(\lnot H)} = \frac{P(\lnot H \land G)}{1-P(H)}$.

But then I got stuck here and didn't know how to rearrange the top.

I later saw the answer, and then when I thought about it using a venn diagram, I realized it made sense. But I'm looking for how you would think about rearranging the top without a venn diagram. Thanks!

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I guess you're almost there. $P(\lnot H \land G)$ is basically $P(H) - P(G \land H)$. Does that make sense? Not H and G means that you are subtracting the intersection of G and H from G. Use a Venn diagram to understand this better. So your answer should be $\frac{P(G) - P(G \land H)}{1-P(H)}$

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