Bayes' Theorem:
This theorem at times is called as inverse probability theorem.
Let us consider any event 'A' of sample space 'S'. This event would have occurred due to the different causes.
Now, let us say that event A is found to have taken place and we have to find out the probability that it has occurred to occurrence of cause, say Ai.
That means we are interested in P(Ai/A). These types of problems are solved by the help of Bayes' theorem.
From total probability theorem we get P(A) = 
This result is called as Bayes' theorem.
Example: A bag 'A' has 2 w and 3 red balls, a bag 'B' contains 4 w and 5 black balls. A bag is randomly selected and a ball is drawn from it. Drawn ball is observed to be white. Find probability that bag 'B' was selected.
Sol: Bag A Bag B
2 w, 3R 4 w, 5B
Let A1 be the event that bag 'A' is selected and A2 be the event that bag B is selected.
P(A1) = P(A2) =1/2
Let 'A' be event that white ball is drawn from selected bag.
=> P(A/A1) = 2/5, P(A/A2) = 4/9
P(A) = P(A1) . P(A/A1) + P(A2) . P(A/A2)
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