<p>The probability of the two events are 0.25 and 0.50. The probability of both happening together is 0.14. Which of the following is the probability of none of the events happening?</p>
Step-by-Step Solution
Key Concept: Use the complement rule: P(neither A nor B) = 1 - P(A ∪ B), where P(A ∪ B) = P(A) + P(B) - P(A ∩ B) for any two events.
<p><strong>Step 1:</strong> Identify given information: P(A) = 0.25, P(B) = 0.50, P(A ∩ B) = 0.14</p><p><strong>Step 2:</strong> Find P(A ∪ B) using the addition rule:<br>P(A ∪ B) = P(A) + P(B) - P(A ∩ B)<br>P(A ∪ B) = 0.25 + 0.50 - 0.14 = 0.61</p><p><strong>Step 3:</strong> Find P(neither A nor B) using the complement:<br>P(A' ∩ B') = 1 - P(A ∪ B)<br>P(A' ∩ B') = 1 - 0.61 = 0.39</p><p><strong>Step 4:</strong> Verify: P(A only) + P(B only) + P(both) + P(neither) = (0.25-0.14) + (0.50-0.14) + 0.14 + 0.39 = 0.11 + 0.36 + 0.14 + 0.39 = 1.00 ✓</p><p>∴ Answer: <strong>0.39</strong></p>
Correct Answer: A