Question

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

Answer 1

Consider : E₁ = 2 white balls transferred from 1 to 2
E₂ = 2 black balls transferred from 1 to 2
E₃ = 1 white and 1 black ball from 1 to 2
A = white ball drawn from urn 2

Now,  P(E₁) = `(""^10C_2)/(""^13C_2), "P"("E"_2) = (""^3C_2)/(""^13C_2), "P"("E"_3) = (""^10C_1 xx ""3C_1)/(""^13C_2)` 

P(A/E₁) = `(""^5C_1)/(""^10C_1)`, P(A/E₂) = `(""^3C_1)/(""^10C_1)`

P(A/E₃) = `(""^4C_1)/(""^10C_1)`

Now, by law of total probability, 

P(A) = P(E₁). P(A/E₁) + P(E₂). P(A/E₂) + P(E₃). P(A/E₃)

= `(""^10C_2)/(""^13C_2) .(""^15C_1)/(""^10C_1) + (""^3C_2)/(""^13C_2) .(""^3C_1)/(""^10C_1) + (""^10C_1 xx ""^3C_1)/(""^13C_2). (""^4C_1)/(""^10C_1) `

= `(59)/(130)`

P(A) = 0.4538