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.
Prove if P(A|B) = P(A|B'), A and B are independent.
-1
$\begingroup$
probability
independence
-
0The definition of independence according to Wikipedia is that P(A and B) = P(A)*P(B). – 2017-02-15
1 Answers
3
$P(A|B) = P(A|B')$
$\frac{P(A\cap B)}{P(B)}=\frac{P(A\cap B')}{P(B')}$ (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)