Prove the following statements about conditional probabilities: For any two events A and B, P[A|B]=1P[¬A|B]. IfAandBare independent, then¬Aand¬Bare independent.For any three eventsA, B,andC,P[ABC]=P[A]P[B|A]P[C|AB]. Solution according to Bayes theorem for two events X, Y, P[XY] = P[X] P[Y|X] consider AB as one event and C as the other so by applying Bayes theorem P[ABC] = P[AB] P[C|AB] Now applying Bayes theorem for P[AB] So P[ABC] = P[A]P[B|A] P[C|AB].

1. Prove the following statements about conditional probabilities:
For any two events A and B, P[A|B]=1P[¬A|B]. IfAandBare independent, then¬Aand¬Bare
independent.For any three eventsA, B,andC,P[ABC]=P[A]P[B|A]P[C|AB].
Solution
according to Bayes theorem for two events X, Y, P[XY] = P[X] P[Y|X]
consider AB as one event and C as the other
so by applying Bayes theorem P[ABC] = P[AB] P[C|AB]
Now applying Bayes theorem for P[AB]
So P[ABC] = P[A]P[B|A] P[C|AB]