I convinced myself that, if $(X,\mathcal{A},\mu)$ is a probability space, $A,B\in \mathcal{A}$, $\mu(A)=3/4$ and $\mu(B)=2/3$, then $\mu(A\cap B)\geq 5/12$.
How do I find an example on a probability space $(X,\mathcal{A},\mu)$ and $A,B\in \mathcal{A}$, so that $\mu(A)=3/4$, $\mu(B)=2/3$, and $\mu(A\cap B)= 5/12$?
You can use the space on 12 elements, where each element is equally likely to be sampled. A visual representation: $A$ are the points marked with circles, $B$ are the points marked with crosses. $$ \overset{1}O \overset{2}O\overset{3}O\overset{4}O\overset{5}\otimes\overset{6}\otimes\overset{7}\otimes\overset{8}\otimes\overset{9}\otimes\overset{10}\times\overset{11}\times\overset{12}\times $$ (recall that $3/4 = 9/12 $ and $2/3 = 8/12$.)