Probability
Binomial Distribution
Grade 12
Question:
<p>A fair coin is tossed \(n\) times. Given that \(P(\text{at least one head}) > \dfrac{9}{10}\), find the minimum value of \(n\).</p>
<p>(1) 1</p>
<p>(2) 2</p>
<p>(3) 3</p>
<p>(4) 4</p>
Step-by-Step Solution
Key Concept: Use the complement rule: P(at least one head) = 1 - P(no heads) = 1 - (1/2)^n. Set up the inequality 1 - (1/2)^n > 9/10 and solve for the minimum n.
<p><strong>Step 1:</strong> Use the complement rule for "at least one head":</p><p>P(at least one head) = 1 - P(all tails) = 1 - (1/2)^n</p><p><strong>Step 2:</strong> Set up the inequality from the given condition:</p><p>1 - (1/2)^n > 9/10</p><p><strong>Step 3:</strong> Simplify:</p><p>-(1/2)^n > 9/10 - 1</p><p>(1/2)^n < 1/10</p><p><strong>Step 4:</strong> Test integer values:</p><p>• n = 3: (1/2)³ = 1/8 ≈ 0.125 > 0.1 ✗</p><p>• n = 4: (1/2)⁴ = 1/16 ≈ 0.0625 < 0.1 ✓</p><p><strong>Step 5:</strong> Verify n = 4:</p><p>P(at least one head) = 1 - 1/16 = 15/16 = 0.9375 > 0.9 ✓</p><p>∴ Minimum value of n = <strong>4</strong></p>
Correct Answer: D