Two random segments overlapping

HardProbability Random Variables~8m

Let $X_1, X_2, X_3, X_4 \sim \text{Uniform}(0,1)$ be independent. Form the segment from $X_1$ to $X_2$ and the segment from $X_3$ to $X_4$ . What is the probability the two segments overlap?

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Complement: disjoint iff one segment is entirely left of the other, i.e. one pair holds the two smallest points.
That pair being the two smallest has probability $2!2!/4! = 1/6$ ; double it, subtract from $1$ .

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0.6667 (± 0.001)

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Two random segments overlapping

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