Let
A = selection of shop A
B = selection of shop B
E = buying of adulterated mustard oil
\(\therefore P(A) = P(B) = \frac12\)
\(P\left(\frac EA\right) = \frac{40}{30 + 40} = \frac{40}{70} = \frac47\)
\(P\left(\frac EB\right) = \frac{60}{50 + 60} = \frac{60}{110} = \frac6{11}\)
(i) Probability of event of selection shop A when given that adulterated mustard oil bought
\(= P\left(\frac AE\right) =\frac{P(\frac EA)P(A)}{P(\frac EA) P(A) + P(\frac{E}{B})P(B)}\)
\(= \cfrac{\frac47 \times \frac12}{\frac47\times\frac12 + \frac6{11}\times \frac12} \)
\(=\cfrac{\frac12\left(\frac47\right)}{\frac12 \left(\frac47 + \frac6{11}\right)} \)
\(= \cfrac{\frac47}{\frac{44+42}{77}}\)
\(= \frac47 \times \frac{77}{86}\)
\(= \frac{2\times11}{43}\)
\( = \frac{22}{43}\)
(ii) Probability of event of selection shop B when given that adulterated mustard oil bought
\(= P\left(\frac BE\right) =\frac{P(\frac EB)P(B)}{P(\frac EA) P(A) + P(\frac{E}{B})P(B)}\)
\(= \cfrac{\frac6{11} \times \frac12}{\frac47\times\frac12 + \frac6{11}\times \frac12} \)
\(= \cfrac{\frac6{11} \times \frac12}{\frac12 \left( \frac47 + \frac6 {11}\right)}\)
\(= \frac{6}{11} \times \frac{77}{86}\)
\( = \frac{3 \times 7}{43}\)
\( = \frac{21}{43}\)