Are two events compatible/incompatible based only on their probability to happen?

Question

Suppose we have two events $A$ and $B$. We know that $\ P(A) = \frac{1}{3}$ and $P(\overline{B}) = \frac{1}{4}$ (meaning $P(B) = \frac{3}{4}$). How can I find out if $A$ and $B$ are compatible or not, using only their probability to happen?

Answer 1

Assume $A,B$ are two non-compatible events, so they don't happen at the same time, i.e. they are disjoint. Of course $A\cup B \subset \Omega$, where $\Omega$ is the whole event space.

Disjointness of $A$ and $B$ (for the "=") and the properties of the probability measure $P$ deliver: $$P(\Omega) \geq P(A\cup B) = P(A)+P(B) = \frac{1}{3} + \frac{3}{4} > 1,$$ yielding a contradiction since it has to hold $P(\Omega)=1$.