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Home » Principal \( P \) = $10,000 - ToelettAPP

Principal \( P \) = $10,000 - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \binom{5}{3} = 10

📌 \frac{10}{35} = \frac{2}{7}

📌 Hence, the probability is $ \boxed{\frac{2}{7}} $.

📌 Question: A data scientist trains a model using sequences of 5 binary features, each being 0 or 1. If each feature is independently set to 1 with probability $ \frac{1}{3} $, what is the probability that exactly 3 of the 5 features are 1?

📌 This is a binomial probability with $ n = 5 $, $ k = 3 $, and success probability $ p = \frac{1}{3} $.

📌 P(X = 3) = \binom{5}{3} \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^2 = 10 \cdot \frac{1}{27} \cdot \frac{4}{9} = 10 \cdot \frac{4}{243} = \frac{40}{243}

📌 Thus, the probability is $ \boxed{\frac{40}{243}} $.

📌 Question: In a biotech lab simulation, 4 identical red enzymes, 3 identical green enzymes, and 2 identical blue enzymes are deployed in a sequence. How many distinct sequences can be formed where all red enzymes are indistinct, green indistinct, and blue indistinct?

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