<p>What is the probability of guessing correctly at least 8 out of 10 answers on a true–false examination?</p>
Step-by-Step Solution
Key Concept: Use binomial probability formula with n=10, p=0.5 (since true-false has equal probability), and find P(X≥8) = P(X=8) + P(X=9) + P(X=10) by summing individual binomial probabilities.
<p><strong>Step 1:</strong> Identify parameters: n=10 trials, p=0.5 (probability of guessing correctly on true-false), need P(X≥8).</p><p><strong>Step 2:</strong> P(X≥8) = P(X=8) + P(X=9) + P(X=10)</p><p><strong>Step 3:</strong> Using binomial formula P(X=k) = C(n,k)p^k(1-p)^(n-k):</p><p>P(X=8) = C(10,8)(0.5)^8(0.5)^2 = 45 × (1/1024) = 45/1024</p><p>P(X=9) = C(10,9)(0.5)^9(0.5)^1 = 10 × (1/1024) = 10/1024</p><p>P(X=10) = C(10,10)(0.5)^10 = 1 × (1/1024) = 1/1024</p><p><strong>Step 4:</strong> P(X≥8) = (45 + 10 + 1)/1024 = 56/1024 = 7/128</p><p>∴ Answer: <strong>7/128</strong></p>
Correct Answer: 7/128