Question

If \[ P(A) = \frac{3}{13}, P(B) = \frac{5}{13}, \text{and} P(A \cap B) = \frac{2}{13}, \] \(\text{then find the value of }\) \( P(B|A) \).

Hint:

The formula for conditional probability is: \[ P(B|A) = \frac{P(A \cap B)}{P(A)}. \] This allows you to find the probability of event \( B \) occurring given that event \( A \) has already occurred.

Answer 1

We are given the following probabilities: \[ P(A) = \frac{3}{13}, P(B) = \frac{5}{13}, P(A \cap B) = \frac{2}{13}. \] The conditional probability \( P(B|A) \) is given by the formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)}. \] Substituting the given values: \[ P(B|A) = \frac{\frac{2}{13}}{\frac{3}{13}} = \frac{2}{3}. \] Conclusion: The value of \( P(B|A) \) is \[ \boxed{\frac{2}{3}}. \]