Conditional probability p(C|A+B)

Question

I'm trying to express conditional probability of $p(C|A + B)$ (probability of $C$, given $A$ or $B$) using $p(C|A)$, $p(C|B)$, and $p(C|AB)$.

Using Venn's diagram I came up with: $$p(C|A + B) = p(C|A) + p(C|B) - p(C|AB).$$

Is this correct? How can I prove this more formally?

Answer 1

This is incorrect. I think you are confusing the identity:

$$P(A+B | C) = P(A|C)+P(B|C)-P(AB|C)$$

It is this result which you would have shown from venn diagrams. To see that this does not hold just let A,B,C be all independent and a contradiction will immediately arise unless they are all of probability 0.