Question

Meera visits only one of the two temples A and B in her locality. Probability that she visits temple A is \( \frac{2}{5} \). If she visits temple A, the probability that she meets her friend is \( \frac{1}{3} \). The probability that she meets her friend, whereas it is \( \frac{2}{7} \) 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. \( \frac{5}{16} \)
• B. \( \frac{3}{16} \)
• C. \( \frac{9}{16} \)
• D. \( \frac{7}{16} \)

Hint:

When calculating conditional probabilities, remember to use Bayes' Theorem to relate the conditional probabilities and the total probability.

Answer 1

The Correct Option is C

We are given the following information: - \( P(E_1) = \frac{2}{5}, P(E_2) = \frac{3}{5} \) (probabilities of visiting temples A and B respectively). - \( P(A | E_1) = \frac{1}{3} \), the probability that Meera meets her friend at temple A, given that she visits temple A. - \( P(A | E_2) = \frac{2}{7} \), the probability that Meera meets her friend at temple B, given that she visits temple B. We need to find \( P(E_2 | A) \), the probability that Meera visits temple B given that she met her friend. Using Bayes’ Theorem: \[ P(E_2 | A) = \frac{P(A | E_2) P(E_2)}{P(A)} \] We know that: \[ P(A) = P(A | E_1) P(E_1) + P(A | E_2) P(E_2) \] Substituting the given values: \[ P(A) = \left(\frac{1}{3} \times \frac{2}{5}\right) + \left(\frac{2}{7} \times \frac{3}{5}\right) \] \[ P(A) = \frac{2}{15} + \frac{6}{35} = \frac{14}{35} = \frac{2}{5} \] Now applying Bayes' Theorem: \[ P(E_2 | A) = \frac{\left(\frac{2}{7} \times \frac{3}{5}\right)}{\frac{9}{16}} = \frac{9}{16} \] Thus, the probability that Meera met her friend at temple B is \( \frac{9}{16} \).