Question

what is bayes theorem in probability

Answer 1

Hi there!

Bayes' theorem: If E1, E2,.....,En are n non empty events which constitute a partition of sample space S, i.e. E1, E2,......,En are pairwise disjoint and E1 U E2 U.........U En = S and A is any event of non zero probability.

Formula is given in the above attachment.
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For better illustration, let's take an example..

Ques.--> Bag A contains 3 red and 4 black balls while another bag B contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag B.

Solution -> Let
• E1 be the event of choosing the bag A,
• E2 the event of choosing the bag B and
• X be the event of drawing a red ball.

Then, P (E1) = P (E2) = 1/2

Also, P (X/E1) = P(drawing a red ball from bag A) = 3/7

and, P (X/E2) = P(drawing a red ball from bag B) = 5/11

Now the probability of drawing a ball from bag B, being given that it is red, is P(E2/X)

By using Bayes'theorem,

P(E2/X) = P(E1) P(X/E2) / P(E1) P(X/E1) + P(E2) P(X/E2)
= (1/2 × 5/11) / ( 1/2 × 3/7 + 1/2 × 5/11)
= 35/68

Hope it helps!