Proof of the law of total probability: Right side = P(A1)P(B|A1) +...+ P(Ak)P(B|Ak) (Reason 1?) = P(B A1) +...+ P(B Ak) (Reason 2?) = P{(B A1) ... (B Ak)} (Reason 3?) = P{B (A1 ... Ak)} (Reason 4?) = P(B S) = P(B) = Left side Proof of Bayes Theorem: P(Aj|B) = P(Aj B)/P(B) Reason 5? P{Aj B) = P(B|Aj)P(Aj) Reason 6 ? P(B) = P(Ai)P(B|Ai) Reason 7? Then combining those formula, we can get P(Aj|B) = P(Aj B)/P(B) = P(Aj)P(B|Aj)/ P(Ai)P(B|Ai)