Probability
Independent Events
Grade 12
Question:
<p>The probability that a 50-year-old man will be alive at 60 is 0.83 and the probability that a 45-year-old woman will be alive at 55 is 0.87. Then</p>
<p>the probability that both will be alive is 0.7221</p>
<p>at least one of them will alive is 0.9779</p>
<p>at least one of them will alive is 0.8230</p>
<p>the probability that both will be alive is 0.6320</p>
Step-by-Step Solution
Key Concept: When two independent events are involved (survival of two different people), multiply their individual probabilities to find the joint probability. Independence is key—one person's survival doesn't affect the other's.
<p><strong>Step 1:</strong> Identify the given probabilities.</p><p>P(50-year-old man alive at 60) = 0.83</p><p>P(45-year-old woman alive at 55) = 0.87</p><p><strong>Step 2:</strong> Recognize these are independent events (one person's survival is independent of the other's).</p><p><strong>Step 3:</strong> For independent events, P(A and B) = P(A) × P(B).</p><p>P(both alive) = 0.83 × 0.87 = 0.7221</p><p><strong>Step 4:</strong> For other standard probabilities:</p><p>P(man dies, woman alive) = (1 - 0.83) × 0.87 = 0.17 × 0.87 = 0.1479</p><p>P(man alive, woman dies) = 0.83 × (1 - 0.87) = 0.83 × 0.13 = 0.1079</p><p>P(both die) = 0.17 × 0.13 = 0.0221</p><p>∴ Answer: A</p>
Correct Answer: A