Question

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

• A. \(\frac{14}{41}\)
• B. \(\frac{13}{31}\)
• C. \(\frac{15}{51}\)
• D. \(\frac{16}{61}\)

Hint:

In conditional probability questions involving reverse probability, use Bayes' theorem and first calculate the total probability of the given condition.

Answer 1

The Correct Option is A

Step 1: Note the premium chances.
Brand shares are $0.40,\,0.35,\,0.25$ and the premium rates inside each brand are $0.30,\,0.40,\,0.60$.

Step 2: Weight each brand by its premium buyers.
The share of all customers who buy premium from each brand is $0.40\times0.30=0.12$, $0.35\times0.40=0.14$, $0.25\times0.60=0.15$.

Step 3: Add for the total premium share.
\[ P(\text{premium})=0.12+0.14+0.15=0.41. \]

Step 4: Take the Brand B slice.
The wanted probability is the Brand B premium piece over the total: \[ P(B\mid \text{premium})=\frac{0.14}{0.41}=\frac{14}{41}. \]

Step 5: Conclude.
\[ \boxed{\frac{14}{41}} \]