Discussion
A retail shop sells three brands of tea, namely Brand A, Brand B, and Brand C, and each in two varieties, namely regular and premium. The proportion of customers buying the Brands A, B, and C are 40%, 35%, and 25%, respectively. Out of those customers who buy Brand A, 30% buy the premium variety, out of those who buy Brand B, 40% buy the premium variety, while out of those who buy Brand C, 60% buy the premium variety. Given that a randomly selected customer has bought the premium variety of tea, the probability that he/she has bought Brand B equals
Answer: A
The probability is \(\frac{14}{41}\).
Let \(Pr\) be the event that the customer buys premium. By the theorem of total probability:
\[ \begin{aligned} P(Pr) &= 0.4(0.3) + 0.35(0.4) + 0.25(0.6) \\ &= 0.12 + 0.14 + 0.15 = 0.41 \end{aligned} \]
Then by Bayes' theorem:
\[ \begin{aligned} P(B \mid Pr) &= \frac{0.35 \times 0.4}{0.41} \\ &= \frac{0.14}{0.41} = \frac{14}{41} \end{aligned} \]
About 34%, a little under the 35% share of Brand B overall, because Brand B's premium rate (40%) is close to the average premium rate (41%). The other fractions do not arise from this data.
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