Given three events A, B, and C such that P (A and B and C) are not equal to 0 and P(C|A and B) = P(C|B), show that P(A| B and C) = P(A|B).
Hello,
$\displaystyle P(C/A\cap B)=\frac{P(A \cap B \cap C)}{P(A \cap B)}=\frac{P(B \cap C)}{P(B)}=P(C/B)$ << this is given.
Hence $\displaystyle \boxed{P(A \cap B \cap C)=\frac{P(A \cap B) \cdot P(B \cap C)}{P(B)}}$
$\displaystyle P(A/ B \cap C)=\frac{P(A \cap B \cap C)}{P(B \cap C)}$
Substitute $\displaystyle P(A\cap B\cap C)$ in this formula, simplify and then recall that $\displaystyle P(A/B)=\frac{P(A \cap B)}{P(B)}$