Probability
Independent Events
Grade 12
Question:
<p>The probability that the wife will be alive 10 years hence is \(\frac{7}{15}\) and that of the husband is \(\frac{7}{10}\). What is the probability that at least one of them will be alive 10 years hence?</p>
<p>(a) \(\frac{21}{150}\)</p>
<p>(b) \(\frac{126}{150}\)</p>
<p>(c) \(\frac{101}{150}\)</p>
<p>(d) \(\frac{94}{150}\)</p>
Step-by-Step Solution
Key Concept: Use the complement rule: P(at least one alive) = 1 - P(both dead). Find P(wife dead) and P(husband dead) independently, then multiply since events are independent.
<p><strong>Step 1:</strong> Identify the given probabilities.</p><p>P(wife alive) = 7/15, so P(wife dead) = 1 - 7/15 = 8/15</p><p>P(husband alive) = 7/10, so P(husband dead) = 1 - 7/10 = 3/10</p><p><strong>Step 2:</strong> Apply complement rule for 'at least one'.</p><p>P(at least one alive) = 1 - P(both dead)</p><p><strong>Step 3:</strong> Calculate P(both dead) using independence.</p><p>P(both dead) = P(wife dead) × P(husband dead) = (8/15) × (3/10) = 24/150 = 4/25</p><p><strong>Step 4:</strong> Find the final probability.</p><p>P(at least one alive) = 1 - 4/25 = 21/25</p><p>∴ Answer: C (21/25)</p>
Correct Answer: C