Probability
Binomial Distribution
Grade 12
Question:
<p>Consider 5 independent Bernoulli's trials each with probability of success \(p\). If the probability of at least one failure is greater than or equal to \(\dfrac{31}{32}\), then \(p\) lies in the interval</p>
<p>\(\left(\dfrac{11}{12}, 1\right]\)</p>
<p>\(\left(\dfrac{1}{2}, \dfrac{3}{4}\right]\)</p>
<p>\(\left[\dfrac{3}{4}, \dfrac{11}{12}\right]\)</p>
<p>\(\left[0, \dfrac{1}{2}\right]\)</p>
Step-by-Step Solution
Key Concept: The probability of at least one failure equals 1 minus the probability of all successes: P(at least one failure) = 1 - p^5. Set up the inequality 1 - p^5 ≥ 31/32 and solve for p.
<p><strong>Step 1:</strong> For 5 independent Bernoulli trials with success probability p, the probability of all 5 successes is p⁵.</p><p><strong>Step 2:</strong> The probability of at least one failure is the complement: P(at least one failure) = 1 - p⁵</p><p><strong>Step 3:</strong> Set up the given inequality: 1 - p⁵ ≥ 31/32</p><p><strong>Step 4:</strong> Rearrange: -p⁵ ≥ 31/32 - 1 = -1/32</p><p><strong>Step 5:</strong> Multiply both sides by -1 (reverse inequality): p⁵ ≤ 1/32</p><p><strong>Step 6:</strong> Take the fifth root: p ≤ (1/32)^(1/5) = (1/2⁵)^(1/5) = 1/2</p><p><strong>Step 7:</strong> Since p is a probability: 0 ≤ p ≤ 1, combined with p ≤ 1/2 gives p ∈ [0, 1/2] or p ∈ (0, 1/2]</p><p>∴ Answer: D</p>
Correct Answer: D