How to derive $P(PH)$ without using a joint distribution table?

[tmt1]

Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?

 

[I like Serena]

Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?

Hi tmt,

We can use that (applying sum rule and general product rule):
$$P(A) = P((A\land B)\lor (A\land \lnot B)) = P(A\land B) + P(A\land\lnot B) = P(A\mid B)P(B) + P(A\mid \lnot B)P(\lnot B)$$