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📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 Evaluate the limit as \(x o 2\):

📌 \[ \lim_{x o 2} (x + 2) = 4 \]

📌 #### 4Question: A physicist analyzing particle collisions observes that in a series of 12 independent trials, each trial results in one of three outcomes: success (S), partial success (P), or failure (F), with probabilities 0.4, 0.3, and 0.3 respectively. What is the probability that exactly 5 successes, 4 partial successes, and 3 failures occur across the 12 trials?

📌 Solution: This is a multinomial probability problem. Let $ X = (X_S, X_P, X_F) $ be the counts of each outcome in 12 independent trials, with probabilities $ p_S = 0.4 $, $ p_P = 0.3 $, and $ p_F = 0.3 $. The probability mass function of a multinomial distribution gives:

📌 P(X_S = 5, X_P = 4, X_F = 3) = rac{12!}{5! \cdot 4! \cdot 3!} \cdot (0.4)^5 \cdot (0.3)^4 \cdot (0.3)^3

📌 First, compute the multinomial coefficient:

📌 rac{12!}{5! \cdot 4! \cdot 3!} = rac{479001600}{120 \cdot 24 \cdot 6} = rac{479001600}{17280} = 27720

📌 Now compute the product of probabilities:

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