How to prove: $P(A\mid B\ \&\ C) = P(A\mid B)+P(A\mid C)-P(A\mid B)P(A\mid C)$ How do we prove:
$$P(A\mid B\ \&\ C) = P(A\mid B)+P(A\mid C)-P(A\mid B)P(A\mid C)$$
if $B$ and $C$ are not mutually exclusive (and $B \neq$ C). Again not a homework question. I am a biologist trying to understand some math. Thanks!

Under what hypothesis are we taking the formula? I don't think it is true in a general context. For instance, if $B=C$, the formula states
$$
P(A|B) = 2P(A|B) - P(A|B)^2
$$
$$
0 = P(A|B)[1-P(A|B)]
$$
which is not true in general.
