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In Probability Bayes’ Theorem is used for calculating conditional probability. Conditional probability can be said as the happening or non-happening of an event under certain given conditions or circumstances. Ex: X says to Y that I’ll buy you coffee, if it rains today. Then, the probability of A buying coffee is dependent on the condition that it rains that very day. So, it is a conditional probability, meaning occurring of event A is dependent on the occurrence of event B.

Bayes’ Theorem was named after Thomas Bayes, a British Mathematician who published the theorem in 1763. It is one of the important theorems used in conditional probabilities. 

The formula for Bayes Theorem is as follows:
 

Problems on Bayes’ Theorem

Example 1
A bag contains 6 black marbles and 4 white marbles. Two marbles are drawn without replacement from the bag. What is the probability that both of the marbles are red?

Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is white. We know the following:

• In the beginning, there are 10 marbles in the bag, 4 of which are white. Therefore, P(A) = 4/10.
• After the first selection, there are 9 marbles in the bag, 3 of which are white. Therefore, P(B|A) = 3/9.

Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15

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