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Because then the earliest and latest arrivals are within 2 minutes, and the entire set fits in [X_{(1)}, X_{(1)} + 2].

February 22, 2026

$Because then the earliest and latest arrivals are within 2 minutes, and the entire set fits in [X_{(1)}, X_{(1)} + 2].$

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)} $.
For all six points to lie within some interval of length 2, it is necessary and sufficient that:
X_{(6)} - X_{(1)} \leq 2
Now, we compute:
P(X_{(6)} - X_{(1)} \leq 2) \quad \text{for } X_i \sim \text{Uniform}[0,10]
Scale: Let $ Y_i = X_i / 10 \sim \text{Uniform}[0,1] $. Then $ Y_{(1)} \leq Y_{(6)} \leq 1 $, and condition becomes $ Y_{(6)} - Y_{(1)} \leq 0.2 $