A short wait for the next train

Question

Trains depart a platform exactly every 12 minutes. You arrive at a uniformly random time, unaware of the schedule. What is the probability you wait less than 4 minutes for the next train?

Accepted Answer

Because the trains are evenly spaced and your arrival is uniform, the time until the next train is uniform on [0, 12] minutes. A wait under 4 minutes covers the interval [0, 4): P(	ext{wait} < 4) = \frac{4}{12} = \frac13 . Takeaway. With evenly spaced departures every T and a uniformly timed arrival, your wait is uniform on [0, T]. Any wait-time probability is just a length over T: here 	frac{4}{12} = 	frac13.

A short wait for the next train

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