I don't know if the next statement is true or false:
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $A,B$ and $C$ be events in $\mathcal{F}$ such that $P(A)>0$. If $P(B|A)>P(B)$ and $P(C|A)>P(C)$, then $P(B\cap C|A)\geq P(B\cap C)$.
Neither I haven't been able to prove it, nor I haven't been able to find a counterexample. Can anyone help to determine the truth or falsity of the statement please?
This is not the case.
Say we uniformly randomly pick an integer from $[1,19]$. Then it has probability $\frac{10}{19}$ to be odd, $\frac8{19}$ to be prime and $\frac7{19}$ to be an odd prime. If I now tell you that it’s in $[1,9]$, the probability that it’s odd increases to $\frac59$ and the probability that it’s prime increases to $\frac49$, but the probability that it’s an odd prime decreases to $\frac39$.