Can we say that $P(A\mid B) < P(A)$?

Question

$P(A\mid B) = \dfrac{P(A \cap B)}{P(B)}$

and we know that $A \cap B \subset A$

So shouldn't $P(A \mid B) < P(A)$ ?

Answer 1

Mistake: $A \cap B < B$ should be $A \cap B \subseteq B$ , and we do have $P(A \cap B) \leq P(B)$.

Hence you can conclude that $P(A|B) \leq 1$.

We can't conclude for sure that $P(A|B) < P(A)$, in particular $A=B$ can be the event that a fair coin gives you tail.

Then $P(A|B)=1 \ge P(A)=\frac12.$