Question

The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

Answer 1

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

(I) By transferring a black ball from bag A to bag B, then drawing a white ball
(II) By transferring a white ball from bag A to bag B, then drawing a white ball

Let E₁, E₂ and A be events as defined below:
E₁ = A black ball is transferred from bag A to bag B
E₂ = A white ball is transferred from bag A to bag B
A = A white ball is drawn

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{7}{15} \\ P\left(E_2\right)=\frac{8}{15} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=\frac{5}{10}=\frac{1}{2} \\ P\left(A/E_2\right)=\frac{6}{10}=\frac{3}{5} \\ \mathrm{Using}\mathrm{the}\mathrm{law}\mathrm{of}\mathrm{total}\mathrm{probability},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(A\right)=P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right) \\ =\frac{7}{15}\times \frac{1}{2}+\frac{8}{15}\times \frac{3}{5} \\ =\frac{7}{30}+\frac{8}{25} \\ =\frac{35+48}{150}=\frac{83}{150} \end{gathered}$$