<p>An experiment succeeds twice as often as it fails. Then find the probability that in the next 6 trials, there will be at least 4 successes.</p>
Step-by-Step Solution
Key Concept: If success is twice as likely as failure, then p = 2/3 and q = 1/3. Use binomial probability for P(X ≥ 4) = P(X=4) + P(X=5) + P(X=6) in 6 trials.
<p><strong>Step 1:</strong> Find individual probabilities. If success occurs twice as often as failure: p = 2q and p + q = 1. Solving: 2q + q = 1 → q = 1/3, p = 2/3.</p><p><strong>Step 2:</strong> Calculate P(X = 4) in 6 trials:</p><p>P(X = 4) = C(6,4)(2/3)⁴(1/3)² = 15 × (16/81) × (1/9) = 240/729</p><p><strong>Step 3:</strong> Calculate P(X = 5):</p><p>P(X = 5) = C(6,5)(2/3)⁵(1/3)¹ = 6 × (32/243) × (1/3) = 192/729</p><p><strong>Step 4:</strong> Calculate P(X = 6):</p><p>P(X = 6) = C(6,6)(2/3)⁶(1/3)⁰ = 1 × (64/729) × 1 = 64/729</p><p><strong>Step 5:</strong> Add all probabilities:</p><p>P(X ≥ 4) = 240/729 + 192/729 + 64/729 = 496/729</p><p>∴ Answer: <strong>496/729</strong></p>
Correct Answer: 496/729