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how to play jeopardy - Sourci

📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 5Question: A materials scientist is testing a composite material with 5 microscopic cracks, each independently healed with probability $ \frac{1}{3} $. What is the probability that exactly 2 cracks are healed?

📌 Solution: The probability follows a binomial distribution with parameters $ n = 5 $ and $ p = \frac{1}{3} $. The probability of exactly 2 successes is $ \binom{5}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^3 $. Calculating:

📌 \binom{5}{2} = 10, \quad \left(\frac{1}{3}\right)^2 = \frac{1}{9}, \quad \left(\frac{2}{3}\right)^3 = \frac{8}{27}.

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