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Home » Not helpful. Try factoring as a quadratic in $ x^2 $: - Groen Casting

Not helpful. Try factoring as a quadratic in $ x^2 $: - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Thus, the number of valid distinct arrangements is:

📌 \boxed{40}

📌 Question: A smart city integrator in San Francisco is analyzing traffic patterns and observes that 6 autonomous vehicles arrive at an intersection at random times during a 10-minute window. Each vehicle independently chooses a 2-minute interval (possibly overlapping) during the window: [0,10), and we are to compute the probability that all 6 vehicles arrive within some continuous 2-minute subinterval of [0,10), assuming uniform and independent arrival times.

📌 Solution: This is a continuous probability problem involving uniform random variables and coverage intervals.

📌 Let each vehicle arrive at a time uniformly distributed over [0,10). We want the probability that there exists a 2-minute interval [t, t+2) ⊆ [0,10) such that all 6 arrival times fall within [t, t+2).

📌 Standard technique: Compute the complementary probability — that **no** such interval covers all 6 points — and subtract from 1. However, due to complexity, we instead use geometric probability and order statistics.

📌 Let $ X_1, X_2, \dots, X_6 \sim \text{Uniform}[0,10] $, and define $ X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(6)} $.

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