Question

A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.

Answer 1

A red ball can be drawn in two mutually exclusive ways:

(I) By transferring a black ball from first to second bag, then drawing a red ball
(II) By transferring a red ball from first to second bag, then drawing a red ball

Let E₁, E₂ and A be the events as defined below:
E₁ = A black ball is transferred from first to second bag
E₂ = A red ball is transferred from first to second bag
A = A red ball is drawn

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{8}{14} \\ P\left(E_2\right)=\frac{6}{14} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=\frac{8}{15} \\ P\left(A/E_2\right)=\frac{9}{15} \\ \mathrm{Using}\mathrm{the}\mathrm{law}\mathrm{of}\mathrm{total}\mathrm{probability},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right) \\ =\frac{8}{14}\times \frac{8}{15}+\frac{6}{14}\times \frac{9}{15} \\ =\frac{64}{210}+\frac{54}{210} \\ =\frac{118}{210}=\frac{59}{105} \end{gathered}$$