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

Question

asked Jul 12, 2020 34.8k views

1 Answer

Mustafa Shabib · Answer 1 · 2020-07-16T15:47:42+0000

By definition of conditional probability,

$P(A\mid B)=(P(A\cap B))/(P(B))$

$P(B\mid A)=(P(A\cap B))/(P(A))$

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

$(P(A\cap B))/(P(B))=(P(A\cap B))/(P(A))\impliesP(A\cap B)\left(\frac1{P(B)}-\frac1{P(A)}\right)=0$

so that either

$P(A\cap B)=0$

which means the events A and B are disjoint, or

$\frac1{P(B)}-\frac1{P(A)}=0\implies P(A)=P(B)$

which means A and B are equally likely to occur.