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 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
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In conditional probability questions involving reverse probability, use Bayes' theorem and first calculate the total probability of the given condition.
Updated On: Jun 4, 2026
- \(\frac{14}{41}\)
- \(\frac{13}{31}\)
- \(\frac{15}{51}\)
- \(\frac{16}{61}\)
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The Correct Option is A
Solution and Explanation
Step 1: Define the events.
Let \(A\), \(B\), and \(C\) denote the events that a customer buys Brand A, Brand B, and Brand C, respectively.
Let \(P\) denote the event that the customer buys the premium variety.
Given,
\[ P(A)=0.40,\quad P(B)=0.35,\quad P(C)=0.25 \] and
\[ P(P|A)=0.30,\quad P(P|B)=0.40,\quad P(P|C)=0.60 \] We need to find
\[ P(B|P) \]
Step 2: Use Bayes' theorem.
By Bayes' theorem,
\[ P(B|P)=\frac{P(P|B)P(B)}{P(P)} \]
So, first we need to calculate the total probability of buying premium variety, that is \(P(P)\).
Step 3: Calculate the total probability of buying premium variety.
Using the law of total probability,
\[ P(P)=P(P|A)P(A)+P(P|B)P(B)+P(P|C)P(C) \]
Substitute the values:
\[ P(P)=0.30(0.40)+0.40(0.35)+0.60(0.25) \]
\[ P(P)=0.12+0.14+0.15 \] \[ P(P)=0.41 \]
Step 4: Calculate the probability of buying Brand B and premium variety.
\[ P(P|B)P(B)=0.40(0.35) \] \[ P(P|B)P(B)=0.14 \]
Step 5: Apply Bayes' theorem.
\[ P(B|P)=\frac{0.14}{0.41} \]
Writing in fraction form,
\[ P(B|P)=\frac{14}{41} \]
Step 6: Final conclusion.
Hence, the probability that the customer bought Brand B given that he/she bought the premium variety is
\[ \boxed{\frac{14}{41}} \]
Let \(A\), \(B\), and \(C\) denote the events that a customer buys Brand A, Brand B, and Brand C, respectively.
Let \(P\) denote the event that the customer buys the premium variety.
Given,
\[ P(A)=0.40,\quad P(B)=0.35,\quad P(C)=0.25 \] and
\[ P(P|A)=0.30,\quad P(P|B)=0.40,\quad P(P|C)=0.60 \] We need to find
\[ P(B|P) \]
Step 2: Use Bayes' theorem.
By Bayes' theorem,
\[ P(B|P)=\frac{P(P|B)P(B)}{P(P)} \]
So, first we need to calculate the total probability of buying premium variety, that is \(P(P)\).
Step 3: Calculate the total probability of buying premium variety.
Using the law of total probability,
\[ P(P)=P(P|A)P(A)+P(P|B)P(B)+P(P|C)P(C) \]
Substitute the values:
\[ P(P)=0.30(0.40)+0.40(0.35)+0.60(0.25) \]
\[ P(P)=0.12+0.14+0.15 \] \[ P(P)=0.41 \]
Step 4: Calculate the probability of buying Brand B and premium variety.
\[ P(P|B)P(B)=0.40(0.35) \] \[ P(P|B)P(B)=0.14 \]
Step 5: Apply Bayes' theorem.
\[ P(B|P)=\frac{0.14}{0.41} \]
Writing in fraction form,
\[ P(B|P)=\frac{14}{41} \]
Step 6: Final conclusion.
Hence, the probability that the customer bought Brand B given that he/she bought the premium variety is
\[ \boxed{\frac{14}{41}} \]
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