Q. 115.0( 1 Vote )

An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two balls are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that it is a white ball.

Answer :

Given:


Urn I contains 10 white and 3 black balls


Urn II contains 3 white and 5 black balls


Two balls are transferred from urn I to urn II and then a ball is drawn from urn II.


There are three mutually exclusive ways to draw a white ball from urn II –


a. Two white balls are transferred from urn I to urn II, and then, a white ball is drawn from urn II


b. Two black balls are transferred from urn I to urn II, and then, a white ball is drawn from urn II


c. A white and a black ball are transferred from urn I to urn II, and then, a white ball is drawn from urn II


Let E1 be the event that two white balls are drawn from urn I, E2 be the event that two black balls are drawn from urn I and E3 be the event that a white and a black ball are drawn from urn I.


Now, we have







We also have







Similarly, we have







Let E4 denote the event that a white is drawn.


Hence, we have





We also have





Similarly, we also have





Using the theorem of total probability, we get


P(E4) = P(E1)P(E4|E1) + P(E2)P(E4|E2) + P(E3)P(E4|E3)






Thus, the probability of the drawn ball being white is.


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