Determine if the events $A$ and $B$ are incompatible if $P(A)=\dfrac{2}{5}$ and $P(B)=\dfrac{5}{7}.$ > Determine if the events $A$ and $B$ are incompatible if $P(A)=\dfrac{2}{5}$ and $P(B)=\dfrac{5}{7}.$ I really don't know how to approach it. To see when two events are compatible or not, we can observe if they have (a) common element(s). Two events are compatible if $A\cap B \neq\emptyset$ and on the contrary, they are incompatible if $A\cap B =\emptyset$.

Inclusion-exclusion tells us that $P(A\cup B) = P(A)+P(B)-P(A\cap B)$

Rearranging, we have $P(A\cap B)=P(A)+P(B)-P(A\cup B)$

We have then $P(A\cap B)=\frac{2}{5}+\frac{5}{7}-P(A\cup B)\geq \frac{2}{5}+\frac{5}{7}-1>0$

Thus, $P(A\cap B)>0$ and so $A\cap B\
eq \emptyset$