Do two guests arrive within five minutes?

Question

Twelve guests each arrive at an independent, uniformly random minute during a $600$ -minute window. Two guests "arrive together" if their arrival minutes are within $5$ of each other.

What is the probability that at least one pair of guests arrives together?

Accepted Answer

Find the chance a single pair arrives together, then use the birthday-problem approximation. Fix one guest's minute. The other arrives together if within 5 minutes on either side, roughly a window of 11 minutes (5 before, the same minute, 5 after) out of 600: p \approx \frac{11}{600} \approx 0.0183. (Guests near the very start or end have slightly fewer neighboring minutes, so this is a small over-estimate, but it is close.) There are \binom{12}{2} = 66 pairs. The chance that no pair arrives together is approximately P(	ext{none together}) \approx e^{-66\,p} = e^{-66 	imes 0.0183} = e^{-1.2…

Do two guests arrive within five minutes?

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