Let A. B. and C be independent events, with P( C) > O.

QUESTION:

Let A. B. and C be independent events, with P( C) > O. Prove that A and B are conditionally independent given G. Solution. We have P(A BIG) = P(AnBnG) n P(G) P(A)P(B)P(G) P(G) = P(A)P(B) = P(A I G)P(B I G), so A and B are conditionally independent given G. In the preceding calculation, the first equality uses the definition of conditional probabilities; the second uses the assumed independence; the fourth uses the independence of A from G, and of B from G.