Question

The given events \( A \) and \( B \) are such that \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), and \( P(A \cap B) = \frac{1}{8} \); then find \( P(A' \cap B') \).

Hint:

Use the inclusion-exclusion principle to find the union of two events, and then use the complement rule to find the desired probability.

Answer 1

Step 1: Using the formula for the probability of the complement: \[ P(A \text{ not} \cap B \text{ not}) = 1 - P(A \cup B). \] Step 2: Use the inclusion-exclusion principle to calculate \( P(A \cup B) \): \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] Substitute the given values: \[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} = \frac{5}{8}. \] Step 3: Now calculate the complement: \[ P(A \text{ not} \cap B \text{ not}) = 1 - \frac{5}{8} = \frac{3}{8}. \] Thus, the answer is \( \frac{3}{8} \).