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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question: A geographer is analyzing 10 satellite images of a coastal region, each showing either a healthy (H) or degraded (D) ecosystem. If each image independently has a 60% chance of showing H and 40% of showing D, what is the probability that exactly 7 images show H?

📌 This is a binomial probability problem with $ n = 10 $, $ k = 7 $, and $ p = 0.6 $.

📌 Question: A museum curator is arranging 7 artifacts: 3 ancient coins, 2 pottery shards, and 2 metal tools. If the curator displays them in a row with indistinguishable items within each category, how many distinct arrangements are possible?

📌 We are arranging a total of 7 items where some are indistinguishable within categories.

📌 This is a permutation of a multiset:

📌 Question: A physicist is analyzing a sequence of 8 quantum measurements, each of which can yield a result of either $ +1 $ or $ -1 $. What is the probability that the sum of all 8 results is exactly zero?

📌 The sum of the 8 results is zero if and only if there are exactly 4 values of $ +1 $ and 4 values of $ -1 $.

📌 The number of favorable sequences is the number of ways to choose 4 positions out of 8 to be $ +1 $ (the rest are $ -1 $):

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