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what is a hex - Rice University

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 Thus, the probability is $ oxed{rac{560}{6561}} $.

📌 Question: A cyclist rides through a city grid and chooses at random one of six possible routes each day, with each route equally likely. Over the course of 10 days, what is the probability that exactly 3 of the chosen routes are scenic, 4 are family-friendly, and 3 are off-road trails, assuming the routes are categorized as follows: 2 scenic, 3 family-friendly, and 1 off-road (with 1 unusual route misclassified as off-road)?

📌 Solution: This is a multinomial distribution with 10 independent trials and 3 categories. Letâs define the counts based on the actual route types:

📌 2 scenic (S), 3 family-friendly (F), 5 off-road (O) â total 10, but only 2 + 3 + 1 = 6 routes exist; however, the problem states the cyclist chooses from 6 routes, 2 S, 3 F, and 1 truly off-road, with the unusual one misclassified or treated as off-road â but since the probabilities are based on route selection, and categories are defined per route, we must interpret:

📌 There are 6 routes: 2 classified as scenic, 3 as family-friendly, 1 as off-road. Each day, the cyclist picks one route uniformly at random (each route has probability $ rac{1}{6} $). So the probability of choosing:

📌 scenic: $ rac{2}{6} = rac{1}{3} $

📌 family-friendly: $ rac{3}{6} = rac{1}{2} $

📌 off-road: $

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