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1, 3, 5, 9, 15, 45 - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 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}.

📌 Multiplying: $ 10 \cdot \frac{1}{9} \cdot \frac{8}{27} = \frac{80}{243} $.

📌 \boxed{\dfrac{80}{243}}

📌 Question: A linguist analyzes a hypothetical language where each word is 4 phonemes long, each phoneme independently from a set of 5. What is the probability that a randomly chosen word has exactly 2 distinct phonemes?

📌 Solution: First, choose 2 phonemes from 5: $ \binom{5}{2} = 10 $. For each pair, the number of 4-phoneme words using both phonemes (but not just one) is $ 2^4 - 2 = 14 $ (subtracting the two monochromatic words). Total favorable outcomes: $ 10 \cdot 14 = 140 $. Total possible words: $ 5^4 = 625 $. Probability:

📌 \frac{140}{625} = \frac{28}{125}.

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