<p>Three students <em>A</em> and <em>B</em> and <em>C</em> are in a swimming race. <em>A</em> and <em>B</em> have the same probability of winning and each is twice as likely to win as <em>C</em>. Find the probability that <em>B</em> or <em>C</em> wins. Assume no two reach the winning point simultaneously.</p>
Step-by-Step Solution
Key Concept: Set up probability equations using the constraint that P(A) = P(B) = 2·P(C), then use the fact that all probabilities sum to 1 to find individual probabilities before calculating P(B or C).
<p><strong>Step 1:</strong> Let P(C wins) = p. Then P(A wins) = 2p and P(B wins) = 2p (given conditions).</p><p><strong>Step 2:</strong> Since one of them must win: P(A) + P(B) + P(C) = 1</p><p>2p + 2p + p = 1</p><p>5p = 1</p><p>p = 1/5</p><p><strong>Step 3:</strong> Therefore: P(C) = 1/5, P(B) = 2/5</p><p><strong>Step 4:</strong> Find P(B or C wins). Since these are mutually exclusive events:</p><p>P(B or C) = P(B) + P(C) = 2/5 + 1/5 = 3/5</p><p>∴ Answer: <strong>3/5</strong></p>
Correct Answer: 3/5