Question

If \( P(A) = \frac{2}{5}, \, P(B) = \frac{1}{4}, \, P(A \cup B) = \frac{1}{2}, \) then \( P(A' \cup B') = \)

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{3}{20} \)
• D. \( \frac{17}{20} \)

Hint:

The complement rule helps to easily find the probability of the union of complements.

Answer 1

The Correct Option is D

Step 1: Use the formula for \( P(A \cup B) \).
We know that \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting the given values, \[ \frac{1}{2} = \frac{2}{5} + \frac{1}{4} - P(A \cap B) \] Step 2: Solve for \( P(A \cap B) \).
\[ P(A \cap B) = \frac{2}{5} + \frac{1}{4} - \frac{1}{2} = \frac{8}{20} + \frac{5}{20} - \frac{10}{20} = \frac{3}{20} \] Step 3: Use the complement rule.
Since \( P(A' \cup B') = 1 - P(A \cup B) \), \[ P(A' \cup B') = 1 - \left(\frac{1}{2}\right) = \frac{1}{2} \] Step 4: Conclusion.
Hence, \[ P(A' \cup B') = \frac{17}{20} \]