Probability
Classical Probability
Grade 12
Question:
<p>If \(p\) is the probability that a man aged \(x\) will die in a year, then the probability that out of \(n\) men \(A_1, A_2, \ldots, A_n\) each aged \(x\), \(A_1\) will die in an year and be the first to die is</p>
<p>\(1 - (1-p)^n\)</p>
<p>\((1-p)^n\)</p>
<p>\(\frac{1}{n}[1-(1-p)^n]\)</p>
<p>\(\frac{1}{n}(1-p)^n\)</p>
Step-by-Step Solution
Key Concept: The probability that A₁ dies first requires A₁ to die AND all other (n-1) men to survive. Since these events are independent, multiply individual probabilities: p × (1-p)^(n-1).
<p><strong>Step 1:</strong> Identify the condition. 'A₁ dies first' means A₁ dies in the year AND A₁ is the only one among the n men who dies.</p><p><strong>Step 2:</strong> A₁ dies in a year with probability <em>p</em>.</p><p><strong>Step 3:</strong> For A₁ to be <strong>first to die</strong>, the other (n-1) men must all survive the year. Each survives with probability (1-p).</p><p><strong>Step 4:</strong> The remaining (n-1) men's survival events are independent, so probability all survive = (1-p) × (1-p) × ... × (1-p) = (1-p)^(n-1).</p><p><strong>Step 5:</strong> By independence, multiply: P(A₁ dies AND all others survive) = p × (1-p)^(n-1).</p><p>∴ Answer: <strong>p(1-p)^(n-1)</strong></p>
Correct Answer: C