Question

Help You are told that P(A|B) = P(B|A). Which statement below must be true

Answer 1

By definition of conditional probability,

$P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$

$P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}$

Since $P(A\mid B)=P(B\mid A)$, we have

$\dfrac{P(A\cap B)}{P(B)}=\dfrac{P(A\cap B)}{P(A)}\impliesP(A\cap B)\left(\dfrac1{P(B)}-\dfrac1{P(A)}\right)=0$

so that either

$P(A\cap B)=0$

which means the events A and B are disjoint, or

$\dfrac1{P(B)}-\dfrac1{P(A)}=0\implies P(A)=P(B)$

which means A and B are equally likely to occur.