Probability
Binomial Distribution
Grade 12
Question:
<p>In a binomial distribution \(B\!\left(b,\, p = \dfrac{1}{4}\right)\), if the probability of at least one success is greater than or equal to \(\dfrac{9}{10}\), 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: Use the complement rule: P(at least one success) = 1 - P(no success) = 1 - (1-p)^n. Set up the inequality and solve for n using logarithms.
<p><strong>Step 1:</strong> We have B(n, p = 1/4). We need P(at least one success) ≥ 9/10.</p><p><strong>Step 2:</strong> Using complement: P(at least one success) = 1 - P(no success) = 1 - (1 - 1/4)^n = 1 - (3/4)^n</p><p><strong>Step 3:</strong> Set up the inequality: 1 - (3/4)^n ≥ 9/10</p><p><strong>Step 4:</strong> Simplify: (3/4)^n ≤ 1/10</p><p><strong>Step 5:</strong> Take logarithm on both sides: n·log(3/4) ≤ log(1/10). Since log(3/4) < 0, dividing reverses the inequality: n ≥ log(1/10)/log(3/4) = log(10)/log(4/3)</p><p><strong>Step 6:</strong> Calculate: log(10)/log(4/3) ≈ 1/0.1249 ≈ 8.0. More precisely, n ≥ 8.008...</p><p><strong>Step 7:</strong> Since n must be an integer, n ≥ 9. Therefore n is greater than 8.</p><p>∴ Answer: A</p>
Correct Answer: A