Question

Box - I contains $5$ red and $2$ blue balls while box - II contains $2$ red and $6$ blue balls. A fair coin is tossed. If it turns up head, a ball is drawn from box - I, else a ball is drawn from box II. Find the probability of each of the following :
i) A red ball is drawn
ii) Ball drawn is from box-I if it is blue

Answer 1

$Box1$ 5 red + 2 blue

$Box2$ 2 red + 6 blue

(i) probability that red ball is drawn:

2 cases: 1) (heads + red from box 1)

2) (tails + red from box 2)

probability $=\frac{1}{2}\times \left(\frac{5}{7}\right)+\frac{1}{2}\left(\frac{2}{6}\right)=\frac{5}{14}+\frac{2}{6}=\Rightarrow \frac{1}{2}\left[\frac{15+14}{21}\right]$

$\Rightarrow \frac{29}{42}$

(ii) Ball is drawn from box $1$ if it is blue

$\frac{\frac{1}{2}\times \frac{2}{7}}{\frac{1}{2}\times \frac{2}{7}+\frac{1}{2}\times \frac{6}{8}}$ $\Rightarrow \frac{\frac{2}{7}}{\frac{2}{7}+\frac{6}{8}}$

$\Rightarrow \frac{\frac{1}{7}}{\frac{1}{7}+\frac{3}{8}}$

$\Rightarrow \frac{8}{29}$