multiple condition probability - one condition is independent of the event

Question

Intuitively I think $P(A|B,C) = P(A|B)$ if $C$ is independent of $A$.

But is there formal proof for this? Or is my intuition wrong?

Answer 1

Your intuition is wrong unfortunately.

Hint: consider throwing a fair coin two times (independently) and the events :

• $A$: On the first throw you get heads
• $C$: On the second throw you get heads
• $B$: On both throws you get heads or on both throws you get tails.