Question

Meera visits only one of the two temples $A$ and $B$ in her locality. Probability that she visits temple $A$ is $\dfrac{2}{5}$. If she visits temple $A$, $\dfrac{1}{3}$ is the probability that she meets her friend, whereas it is $\dfrac{2}{7}$ if she visits temple $B$. Meera met her friend at one of the two temples. The probability that she met her friend at temple $B$ is:

• A. $\frac{7}{16}$
• B. $\frac{5}{16}$
• C. $\frac{3}{16}$
• D. $\frac{9}{16}$

Hint:

When using Bayes' Theorem, think of it as "Specific Case Total Cases." Here, it is (Probability of friend at B) divided by (Probability of friend at A + Probability of friend at B).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem is a classic application of Bayes' Theorem. We are given the prior probabilities of visiting each temple and the conditional probabilities of meeting a friend. We need to find the posterior probability of visiting Temple B, given that the event "meeting a friend" has already occurred.

Step 2: Key Formula or Approach:
Bayes' Theorem: $P(B \mid F) = \frac{P(B) \cdot P(F \mid B)}{P(A) \cdot P(F \mid A) + P(B) \cdot P(F \mid B)}$

Step 3: Detailed Explanation:
1. Define the events: - $A$: Visits Temple A, $B$: Visits Temple B, $F$: Meets her friend. 2. Given probabilities: - $P(A) = 2/5$. - Since she visits only one temple, $P(B) = 1 - 2/5 = 3/5$. - $P(F \mid A) = 1/3$ (Meeting friend at A). - $P(F \mid B) = 2/7$ (Meeting friend at B). 3. Calculate Total Probability of meeting her friend $P(F)$: \[ P(F) = [P(A) \cdot P(F \mid A)] + [P(B) \cdot P(F \mid B)] \] \[ P(F) = \left( \frac{2}{5} \cdot \frac{1}{3} \right) + \left( \frac{3}{5} \cdot \frac{2}{7} \right) = \frac{2}{15} + \frac{6}{35} \] Find common denominator (105): \[ P(F) = \frac{14}{105} + \frac{18}{105} = \frac{32}{105} \] 4. Apply Bayes' Theorem for Temple B: \[ P(B \mid F) = \frac{P(B \cap F)}{P(F)} = \frac{18/105}{32/105} \] \[ P(B \mid F) = \frac{18}{32} = \frac{9}{16} \]

Step 4: Final Answer
The probability that she met her friend at temple B is $\frac{9}{16}$.