Q18 of 249 Page 32

Three urns A,B and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.

Given:


Urn A has 6 red and 4 white balls


Urn B has 2 red and 6 white balls


Urn C has 1 red and 5 white balls


Let us assume U1, U2, U3 and A be the events as follows:


U1 = choosing Urn A


U2 = choosing Urn B


U3 = choosing Urn C


A = choosing red ball from urn


We know that each urn is most likely to choose. So, probability of choosing a urn will be same for every Urn.


⇒


⇒


⇒


The Probability of choosing balls from each Urn differs from Urn to Urn and the probabilities are as follows:


⇒ P(A|U1) = P(Choosing red ball from Urn A)


⇒


⇒


⇒


⇒ P(A|U2) = P(Choosing red ball from Urn B)


⇒


⇒


⇒


⇒ P(A|U3) = P(Choosing red ball from Urn C)


⇒


⇒


⇒


Now we find


P(U1|A) = P(The red ball is from Urn A)


Using Baye’s theorem:


⇒


⇒


⇒


⇒


⇒


∴ The required probability is .


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