Prove if P(A|B) = P(A|B'), A and B are independent.

Question

Basically, prove the intuitive definition of independence: if the probability of A such that B has occurred is the same as if it has not occurred, A and B are independent.

Answer 1

$P(A|B) = P(A|B')$

$\frac{P(A\cap B)}{P(B)}=\frac{P(A\cap B')}{P(B')}$ (definition of conditional probability)

$\frac{P(A\cap B)}{P(B)}=\frac{P(A)-P(A\cap B)}{1-P(B)}$ (use Venn diagram to check)

$P(A\cap B)-P(B)P(A\cap B)=P(A)P(B)-P(B)P(A\cap B)$ (cross multiply)

$P(A\cap B)=P(A)P(B)$ (definition of independence of events)