Probability
Binomial Distribution
Grade None
Question:
<p>In a binomial distribution \(B(n,\ p = 1/4)\), if the probability of at least one success is greater than or equal to \(9/10\), then \(n\) is greater than</p>
<p>\(\dfrac{1}{\log_{10} 4 - \log_{10} 3}\)</p>
<p>\(\dfrac{1}{\log_{10} 4 + \log_{10} 3}\)</p>
<p>\(\dfrac{9}{\log_{10} 4 - \log_{10} 3}\)</p>
<p>\(\dfrac{4}{\log_{10} 4 - \log_{10} 3}\)</p>
Step-by-Step Solution
Key Concept: The probability of at least one success equals 1 minus the probability of zero successes: P(X ≥ 1) = 1 - (1-p)^n. Set up the inequality 1 - (3/4)^n ≥ 9/10 and solve for n using logarithms.
<p><strong>Step 1:</strong> Use complement: P(at least one success) = 1 - P(no success)</p><p>P(X ≥ 1) = 1 - P(X = 0) = 1 - (1-p)^n = 1 - (3/4)^n</p><p><strong>Step 2:</strong> Set up the inequality</p><p>1 - (3/4)^n ≥ 9/10</p><p>(3/4)^n ≤ 1/10</p><p><strong>Step 3:</strong> Take logarithm (base preserves inequality since 3/4 < 1)</p><p>n·log(3/4) ≤ log(1/10)</p><p>n·[log(3) - log(4)] ≤ -log(10)</p><p>n·[-0.288] ≤ -1</p><p>n ≥ 1/0.288 ≈ 3.47</p><p><strong>Step 4:</strong> More precisely, using log₁₀: n·log(3/4) ≤ log(0.1)</p><p>n ≥ log(10)/log(4/3) = 1/log(4/3) ≈ 1/0.1249 ≈ 8.01</p><p>∴ n > 8 (or n ≥ 9 depending on options)</p>
Correct Answer: A