<p>In a game called "odd man out" \(m\) (\(m > 2\)) persons toss a coin to determine who will buy refreshments for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. The probability that there is a loser in any game is</p>
Step-by-Step Solution
Key Concept: For there to be an 'odd man out,' exactly one person must get a different outcome (H or T) from all others. This happens when either 1 person gets H and (m-1) get T, or 1 person gets T and (m-1) get H.
<p><strong>Step 1:</strong> For a loser to exist, exactly one person must have a different outcome from the rest.</p><p><strong>Step 2:</strong> Case 1: Exactly 1 person gets H, rest (m-1) get T. Number of ways = C(m,1) = m. Probability = m · (1/2)^m</p><p><strong>Step 3:</strong> Case 2: Exactly 1 person gets T, rest (m-1) get H. Number of ways = C(m,1) = m. Probability = m · (1/2)^m</p><p><strong>Step 4:</strong> Total probability = m · (1/2)^m + m · (1/2)^m = 2m · (1/2)^m = 2m/2^m = <strong>m/2^(m-1)</strong></p><p>∴ Answer: B (or equivalent form: m·2^(1-m))</p>
Correct Answer: B