Bayes’ Theorem is a mathematical formula that deals with conditional probabilities. That is, it provides a means for updating probabilities based on relevant evidence that has occurred. Also, the theorem is useful when trying to solve a probability problem that seems intractable if not unsolvable at first pass. For example, suppose you find yourself standing outside of three darkened rooms labeled A, B, and C, and you know that in each room there are two people, either male (MM), female (FF), or one of each (MF). Therefore, P(M|A)=1.0; P(M|B)=0.5 and P(M|C)=0. See the following diagram:
You take a flashlight and enter a room at random where you first shine the light on a male. What is the probability that you have entered room A? In other words, find P(A|M).
Seeing that we know the probability rule where P(M and A)=P(M∩A)=P(M)P(A│M), solving for P(A│M), we get
but here, we are stuck in a circular loop! We need to use Bayes’ Theorem to solve the problem:
since commutability holds where 
Hence, from here,
represents the Bayes’ Theorem. So,
Bayes’ Theorem can be generalized to:
