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Question

State and prove Bayes' theorem

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Solution

The probability of two events A and B happening, P(AB), is the probability of A (P(A)),times the probability of B given that A has occurred (P(B|A)).
P(AB)=P(A)P(B|A) ....(1)
On the other hand, the probability of A and B is also equal to the probability of B times the probability of A given B.
P(AB)=P(B)P(A|B) ....(2)
Equating the two yields:
P(B)P(A|B)=P(A)P(B|A) .....(3)
and thus
P(A|B)=P(A)P(B|A)P(B) ....(4)
This equation, known as Bayes' theorem is the basis of statistical inference.

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