use the law of total probabilty to prove that
a. if P (A l B) = P (A l B,) then A and B are independent
b. if P (A l C ) > P (B l C) and P (A l C' ) > P( B l C' ), then P (A) > P (B)
Hello,
The law of total probability says that $\displaystyle P(A)=P(A|B)P(B)+P(A|B')P(B')$
So here, $\displaystyle P(A)=P(A|B)[P(B)+P(B')]=P(A|B)$ since B and B' are complementary events.
Provided that $\displaystyle P(B)\neq 0$, it follows that $\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}=P(A) \Rightarrow P(A\cap B)=P(A)P(B)$
For the second one... :
$\displaystyle P(A)=P(A|C)P(C)+P(A|C')P(C')$
But from the two inequalities, we can say that $\displaystyle P(A)>P(B|C)P(C)+P(B|C')P(C')$, and the RHS is exactly $\displaystyle P(B)$