Question

In a shop X, $30$ tin pure ghee and $40$ tin adultered ghee are kept for sale while in shop Y, $50$ tin pure ghee and $60$ tin adultered ghee are there. One tin of ghee is purchased from one of the shops randomly and it is found to be adultered. Find the probability that it is purchased from shop B.

Answer 1

Given In shop X pure ghee$=30$ tin In shop Y pure ghee$=50$ tin

Adultered ghee$=40$ Adultered ghee$=60$

$P\left(A/B\right)=\frac{P\left(B/A\right)\left(A\right)}{P\left(B\right)}$

$A\to$ the tin is purchased from shop B

$B\to$ the tin is adultered

$P\left(B/A\right)=$probability of the tin from shop 'B' being adultered

$P\left(B/A\right)=\frac{60}{60+50}=\frac{60}{110}=\frac{6}{11}$

$P\left(A\right)=$ Probability that the tin is from shop B

$P\left(A\right)=\frac{(50+60)}{(50+60)+(30+40)}=\frac{110}{180}=\frac{11}{18}$

$P\left(B\right)=$ probability that the tin is adultered

$P\left(B\right)=\frac{(60+40)}{(50+60)+(30+40)}=\frac{100}{180}=\frac{10}{18}$

$P\left(A/B\right)=$ probability that the tin is from shop B given it is adultered

$=\frac{P\left(B/A\right)\left(P\right(A\left)\right)}{P\left(B\right)}=\frac{\frac{6}{11}\times \frac{11}{18}}{\left(\frac{10}{18}\right)}=\frac{6}{10}=\frac{3}{5}$

$\therefore$ probability that the adultered tin is from shop B is $\frac{3}{5}$.