Probability of an event conditional on union of two events

Question

Let $A,B,C$ be three events such that $A,B$ are independent and $A,C$ are not independent.

Now I am wondering if $P(A\mid B\cup C) = P(A\mid C)$?

First of all, I think we cannot say $B$ is independent of $C$. But that may not be relevant to the question. I am not sure how to prove/disprove the above statement. Any ideas? Thanks.

Answer 1

Usually you need to have a sense to determine the direction. To disprove this you will need to construct a counter example.

Consider rolling a fair six-sided dice.

Let $A$ be the event of rolling odd number, $B$ be the event of rolling $1$ or $2$, $C$ be the event of rolling $4$, $5$, or $6$

Then $P(A) = 3/6 = 1/2$, $P(A|B) = 1/2 = P(A)$ so $A, B$ are independent. $P(A|C) = 1/3 \neq 1/2 = P(A)$ so $A, C$ are dependent

So these events $A, B, C$ satisfy your conditions.

But $P(A|B\cup C) = 2/5 \neq 1/3 = P(A|C)$, so this is a counter example disproving the statement.