<p>The probability that a student is not a swimmer is 1/5. Then find the probability that out of 5 students exactly 4 are swimmers.</p>
Step-by-Step Solution
Key Concept: This is a binomial probability problem where we need P(X = 4) with n = 5 trials, probability of success (swimmer) p = 4/5, and we want exactly 4 successes. Use the binomial formula: C(n,k) × p^k × (1-p)^(n-k).
<p><strong>Step 1:</strong> Identify the given information.</p><p>P(not a swimmer) = 1/5</p><p>∴ P(swimmer) = 1 - 1/5 = 4/5</p><p>We need exactly 4 swimmers out of 5 students.</p><p><strong>Step 2:</strong> Apply the binomial probability formula.</p><p>P(X = 4) = C(5,4) × (4/5)^4 × (1/5)^(5-4)</p><p><strong>Step 3:</strong> Calculate the binomial coefficient.</p><p>C(5,4) = 5!/(4!×1!) = 5</p><p><strong>Step 4:</strong> Substitute and simplify.</p><p>P(X = 4) = 5 × (4/5)^4 × (1/5)^1</p><p>= 5 × (4/5)^4 × (1/5)</p><p>= (4/5)^4 × 5 × (1/5)</p><p>= (4/5)^4 × 1</p><p>∴ Answer: $\left(\dfrac{4}{5}\right)^4$</p>
Correct Answer: \(\left(\dfrac{4}{5}\right)^4\)