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

To determine the probability that Meera met her friend at temple \(B\), we can use the concept of conditional probability and the law of total probability.

1. The probability that Meera visits temple \(A\) is \(\frac{2}{5}\), and the probability that she visits temple \(B\) is \(\frac{3}{5}\) (since she only visits one of the two).
2. If Meera visits temple \(A\), the probability that she meets her friend is \(\frac{1}{3}\).
3. If she visits temple \(B\), the probability that she meets her friend is \(\frac{2}{7}\).
4. Using the law of total probability, the probability that Meera meets her friend is given by:

\(P(\text{Meets friend}) = P(\text{Meets friend | Visits A}) \cdot P(\text{Visits A}) + P(\text{Meets friend | Visits B}) \cdot P(\text{Visits B})\)

Substitute the known probabilities:

\(= \left(\frac{1}{3} \cdot \frac{2}{5}\right) + \left(\frac{2}{7} \cdot \frac{3}{5}\right)\)

Calculate each term:

\(= \frac{2}{15} + \frac{6}{35}\)

To add these, find a common denominator (which is 105):

\(= \left(\frac{2}{15} \times \frac{7}{7}\right) + \left(\frac{6}{35} \times \frac{3}{3}\right)\)

\(= \frac{14}{105} + \frac{18}{105} = \frac{32}{105}\)

1. We want the probability that Meera met her friend at temple \(B\), given that she did meet her friend.

Using Bayes' theorem:

\(P(\text{Temple B | Meets friend}) = \frac{P(\text{Meets friend | Visits B}) \cdot P(\text{Visits B})}{P(\text{Meets friend})}\)

Substitute the known probabilities:

\(= \frac{\frac{6}{35}}{\frac{32}{105}}\)

Simplify:

\(= \frac{6 \times 105}{35 \times 32}\)

\(= \frac{630}{1120}\)

Simplify the fraction:

\(= \frac{9}{16}\)

Hence, the correct answer is \(\frac{9}{16}\).