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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)=(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.