Q:

Probability of $A \cap B$

If $P(A) = P(B) = p$, then $P(A \cap B)$ must be less than $p^2$ or more than $p^2$ or cannot determine ?

My attempt is that $P(A \cap B)$ is always less than or equal to $P(A)*P(B)$ given that the two $A$ and $B$ are independent BUT here in question no relation has been given regarding events $A$ and $B$ apart from their Probabilities being equal .
Can we still determine whether $P(A \cap B)$ will be less than p$^2$ ?

A:

If $A=B$, then $Pr(A \cap B)=Pr(A)=p\ge p^2$
Suppose $0<p$ and if $A$ and $B$ are mutually exclusive, then $Pr(A \cap B)= 0 < p^2$.
As you can see, we can't tell if it is bigger or smaller than $p^2$.

