Do all three arrivals fall within a short window?

Question

Three events happen at independent, uniformly random times over a $1$ -hour period ( $[0, 1]$ ).

What is the probability that all three occur within some $w$ -hour window (equivalently, that the range, last minus first, is at most $w$ )? Then evaluate at $w = 0.5$ .

Accepted Answer

We want P(	ext{range} \le w) for n = 3 uniform points. A clean way: condition on the first (smallest) of the three points sitting at position t, and require the other two to land in [t, t + w]. The density of the minimum of 3 uniforms at t is 3(1 - t)^2 (the min is at t if one of the three is at t and the other two are above t). Given the min is at t, each of the other two points is uniform on [t, 1], and both must fall within w of t. For t \le 1 - w, each lands in range with probability \frac{w}{1 - t}; both do with probability \left(\frac{w}{1-t}ight)^2. P(	ext{range} \le w) = \int0^{1-…

Do all three arrivals fall within a short window?

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