Trains every ten minutes, yet nine visits to one

Question

A commuter has two friends who live in opposite directions from his station. Trains to each friend leave every ten minutes, from the same platform. He arrives at the station at a random time and always boards whichever train leaves first.

After months of this he notices that he has visited the first friend nine times as often as the second.

How can this be, if both lines have the same frequency? Give a precise timetable that produces exactly nine to one.

Accepted Answer

The trains to friend A leave at :00, :10, :20, and so on; the trains to friend B leave one minute later, at :01, :11, :21, and so on. Why that gives nine to one. Think about which train comes next, as a function of the commuter's arrival time within a ten-minute cycle. - If he arrives between :01 and :10 (a nine-minute window), the next departure is the A train at :10. - If he arrives between :00 and :01 (a one-minute window), the next departure is the B train at :01. Arriving at a uniformly random time, he lands in the nine-minute window 90 per cent of the time and the one-minute window 10 p…

Trains every ten minutes, yet nine visits to one

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