Probability
Classical definition of probability
Grade 12
Question:
<p>If \(p\) and \(q\) are chosen randomly from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) with replacement, then determine the probability that the roots of the equation \(x^2 + px + q = 0\) are real.</p>
<p>\(\dfrac{31}{100}\)</p>
<p>\(\dfrac{37}{100}\)</p>
<p>\(\dfrac{41}{100}\)</p>
<p>\(\dfrac{27}{100}\)</p>
Step-by-Step Solution
Key Concept: For real roots, the discriminant Δ = p² - 4q ≥ 0, so q ≤ p²/4. Count valid (p,q) pairs where both are chosen from {1,2,...,10} with replacement, then divide by total outcomes 10².
<p><strong>Step 1:</strong> For real roots, we need Δ = p² - 4q ≥ 0, which means q ≤ p²/4.</p><p><strong>Step 2:</strong> Count valid (p,q) pairs for each p ∈ {1,2,...,10}:</p><ul><li>p=1: q ≤ 0.25 → 0 pairs</li><li>p=2: q ≤ 1 → 1 pair (q=1)</li><li>p=3: q ≤ 2.25 → 2 pairs (q=1,2)</li><li>p=4: q ≤ 4 → 4 pairs (q=1,2,3,4)</li><li>p=5: q ≤ 6.25 → 6 pairs (q=1,2,3,4,5,6)</li><li>p=6: q ≤ 9 → 9 pairs (q=1,...,9)</li><li>p=7: q ≤ 12.25 → 10 pairs (q=1,...,10)</li><li>p=8,9,10: → 10 pairs each</li></ul><p><strong>Step 3:</strong> Total valid pairs = 0 + 1 + 2 + 4 + 6 + 9 + 10 + 10 + 10 + 10 = 62</p><p><strong>Step 4:</strong> Total possible outcomes = 10 × 10 = 100</p><p>∴ Probability = 62/100 = <strong>31/50</strong></p>
Correct Answer: B