Question

If P(A/B) = P(B\A) Then Prove that P(A) - = P(B)​

Answer 1

Answer:

This statement will only be true in the event that P(A) = P(B) since by Bayes’ Theorem we have that

P(A|B)=P(A)P(B|A)P(B)=P(B|A)

if and only if P(A) = P(B).

However, like Justin Rising points out, we have to also consider the event that A and B are mutually exclusive, that is the event that A∩B=∅. Assuming this is the case, and that the events A and B both have probability measure larger than zero, we then have

P(A|B)=P(A∩B)P(B)=0P(B)=P(B∩A)P(A)=P(B|A).

So summing it up, we have that if

P(A)=P(B) or P(A∩B)=0 ⇒ P(A|B)=P(B|A).