Probability
Binomial Distribution
Grade 12
Question:
<p>An experiment succeeds twice as often as it fails. The probability of at least five successes in the six trials of this experiment is</p>
<p>\(\dfrac{496}{729}\)</p>
<p>\(\dfrac{192}{729}\)</p>
<p>\(\dfrac{240}{729}\)</p>
<p>\(\dfrac{256}{729}\)</p>
Step-by-Step Solution
Key Concept: If success is twice as likely as failure, then P(success) = 2/3 and P(failure) = 1/3. Use binomial probability to find P(X ≥ 5) = P(X=5) + P(X=6) for n=6 trials.
<p><strong>Step 1:</strong> Set up the probability. If success is twice as likely as failure: P(S) = 2P(F). Since P(S) + P(F) = 1, we get 2P(F) + P(F) = 1, so P(F) = 1/3 and P(S) = 2/3.</p><p><strong>Step 2:</strong> Find P(at least 5 successes in 6 trials) = P(X=5) + P(X=6) where X ~ Binomial(n=6, p=2/3).</p><p><strong>Step 3:</strong> 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> P(X=6) = C(6,6)(2/3)⁶(1/3)⁰ = 1 · (64/729) · 1 = 64/729</p><p><strong>Step 5:</strong> P(X ≥ 5) = 192/729 + 64/729 = 256/729</p><p>∴ Answer: D</p>
Correct Answer: D