A very short or very long metro wait

Question

A metro train departs exactly every 8 minutes. You arrive at a uniformly random time. What is the probability you wait less than 2 minutes or more than 7 minutes?

Accepted Answer

Your wait is uniform on [0, 8] minutes. The two favorable pieces do not overlap: - less than 2 minutes: interval [0, 2), length 2; - more than 7 minutes: interval (7, 8], length 1. Adding the lengths and dividing by the cycle: P = \frac{2 + 1}{8} = \frac{3}{8} . Takeaway. Disjoint wait conditions add up: total favorable length over T. Here 	frac{2 + 1}{8} = 	frac38.

A very short or very long metro wait

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