<p>A box contains tickets numbered from 1 to 20. Three tickets are drawn from the box with replacement. The probability that the largest number on the tickets is 7 is</p>
Step-by-Step Solution
Key Concept: With replacement, the largest of three draws equals 7 iff all three numbers are ≤7 AND at least one equals 7. This equals P(all ≤7) - P(all ≤6).
<p><strong>Step 1:</strong> Identify the condition for largest number to be 7.</p><p>The largest of three numbers equals 7 when: all three numbers are from {1,2,3,4,5,6,7} AND at least one number is 7.</p><p><strong>Step 2:</strong> Use complementary counting.</p><p>P(max = 7) = P(all ≤ 7) - P(all ≤ 6)</p><p><strong>Step 3:</strong> Calculate P(all ≤ 7).</p><p>Since we draw with replacement: P(all ≤ 7) = (7/20)³ = 343/8000</p><p><strong>Step 4:</strong> Calculate P(all ≤ 6).</p><p>P(all ≤ 6) = (6/20)³ = (3/10)³ = 27/1000 = 216/8000</p><p><strong>Step 5:</strong> Find the probability.</p><p>P(max = 7) = 343/8000 - 216/8000 = 127/8000</p><p>∴ Answer: D</p>
Correct Answer: D