Probability conditioned on disjunction of events

Question

This seems elementary, but I cannot see a quick proof:

For events $A, B, C$ we have $$ P(A \mid B \cup C) \leq \max(P(A \mid B), P(A \mid C)) $$

Is this right?

Answer 1

It is incorrect. Consider this counter-example: $A=\{1,2\}$, $B=\{1,3\}$, $C=\{2,3\}$ and $P(\{1\})=P(\{2\})=P(\{3\})=\frac{1}{3}$.