Question

$P(A\cup B)=P(A\cap B)$ if and only if the ralation between $P\left(A\right)$ and $P\left(B\right)$ is $P\left(A\right)+P\left(B\right)=2P\left(AB\right)$. If this is true enter 1, else enter 0.

Answer 1

$P(A\cup B)=P(A\cap B)$ iff $P\left(A\right)+P\left(B\right)-P(A\cap B)=P(A\cap B)$
iff $P\left(A\right)+P\left(B\right)=2P(A\cap B)$ iff $P\left(A\right)+P\left(B\right)=2P\left(A\right)P\left(B/A\right)=2P\left(AB\right)$
which is the required relation
Note that we may write $P\left(B\right)P\left(A/B\right)$ for $P(A\cap B)$ instead of $P\left(A\right)P\left(B/A\right).$