There are three urns containing 2 white and 3 black balls, 3 white and 2 black balls, and 4 white and 1 black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.

There are 3 urns U1, U2 and U3

U1 = 2 white and 3 black balls


U2 = 3 white and 2 black balls


U3 = 4 white and 1 black balls


Total balls = 5


As there is an equal probability of each urn being chosen



Let E1, E2 and E3 be the event that a ball is chosen from an urn U1,


U2 and U3 respectively.



Now, let A be the event that white ball is drawn.


P(A|E1) is the probability that white ball is chosen from urn U1


P(A|E2) is the probability that white ball is chosen from urn U2


P(A|E3) is the probability that white ball is chosen from urn U3



Now, we have to find the probability that the ball is drawn was from


U2.


We use Bayes’ theorem to find the probability of occurrence of an event A when event B has already occurred.


∴


P(E2|A) is the probability that white ball is selected from urn U2.








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