Question

Bag I contains 3 red and 4 black balls, Bag II contains 5 red and 2 black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. Find the probability that the drawn ball is red in colour.

Answer 1

Let, E₁, E₂, E₃ and A be event 

E₁ = Both transferred ball from Bag I to bag II are red.

E₂ = Both transferred ball from bag I to bag II are black

E₃ = out of two transferred ball one is red and other is black.

Now, `P(E_1) = (""^3C_1)/(""^7C_2) = 1/7`

`P(E_2) = (""^4C_2)/(""^7C_2)= 2/7`

`P(E_3) = (""^3C_1xx^4C_1)/(""^7C_2) = 4/7`

Required probability

`P(E_2/A) = (P(E_2) xx P(A/E_2))/(P(E_1) * P(A/E_1) + P(E_2)) xx P(A/E_2) + P(E_3) xx P(A/E_3)`

`P(A/E_1) = 7/9, P(A/E_2) = 2/9, P(A/E_3) = 6/9`

Now, `(2/7 xx 2/9)/(1/7 xx 7/9 + 2/7 xx 2/9 + 4/7 xx 6/9)`

= `4/35`