simple conditional probability question (3)

Question

Is that true that for any events $A, B, C$, the following holds?

$$P(A|B\cap C)\leq P(A|B)$$

I think the answer is no. Intuitively, adding more condition may cause the probability increase. Is that right?

Answer 1

You are right.

Consider the events:

$A $: today it will rain;

$B $: I am a boy;

$C $: the sky is full of gray clouds.

That inequality does not seem right now.

Answer 2

Assume $0<P(A|B)<1$ and $C= A$. Then $P(A|B) < P(A|B\cap C)=P(A|B \cap A) =1$