<p>A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is</p>
Step-by-Step Solution
Key Concept: When two players independently select from the same set, the probability they DON'T match equals 1 minus the probability they DO match. Since A picks any number and B must avoid A's choice, P(no match) = 24/25.
<p><strong>Step 1:</strong> Identify the sample space. A selects any number from 1 to 25, and B independently selects any number from 1 to 25. Total possible outcomes = 25 × 25 = 625.</p><p><strong>Step 2:</strong> Find probability they WIN (numbers match). If A selects a number, B must select the same number. There are exactly 25 matching pairs: (1,1), (2,2), ..., (25,25). So P(match) = 25/625 = 1/25.</p><p><strong>Step 3:</strong> Find probability they DON'T WIN using complement rule. P(no match) = 1 - P(match) = 1 - 1/25 = 24/25.</p><p><strong>Alternate approach:</strong> A selects any number (probability = 1). For B not to match, B must avoid A's number: probability = 24/25.</p><p>∴ Answer: <strong>24/25</strong></p>
Correct Answer: B