<p>One ticket is selected at random from 100 tickets numbered 00, 01, 02, ..., 98, 99. If \(x_1\) and \(x_2\) denotes the sum and product of the digits on the tickets, then \(P(x_1 = 9/x_2 = 0)\) is equal to</p>
Step-by-Step Solution
Key Concept: Use conditional probability formula: P(A|B) = P(A∩B)/P(B). First identify all tickets where x₂ = 0 (product = 0, meaning at least one digit is 0), then count how many of those have x₁ = 9 (sum = 9).
<p><strong>Step 1:</strong> Find all tickets where x₂ = 0 (product of digits = 0). This requires at least one digit to be 0.</p><p>Tickets with at least one 0: 00, 01, 02, ..., 09 (10 tickets) + 10, 20, 30, ..., 90 (9 tickets) = 19 tickets total.</p><p><strong>Step 2:</strong> Among these 19 tickets, find which have x₁ = 9 (sum of digits = 9).</p><p>From {00, 01, ..., 09}: only 09 has sum = 9 ✓</p><p>From {10, 20, 30, ..., 90}: we need sum = 9, so 90 has sum = 9 ✓</p><p>Total tickets with x₁ = 9 AND x₂ = 0: {09, 90} = 2 tickets.</p><p><strong>Step 3:</strong> Apply conditional probability formula:</p><p>P(x₁ = 9 | x₂ = 0) = (Number of favorable outcomes)/(Total outcomes where x₂ = 0) = 2/19</p><p>∴ Answer: A (2/19)</p>
Correct Answer: A