# Tale of Two Trains

There are two trains that run between two cities. The trains are identical and run on identical routes, so passengers have no preference between the two and would take whichever train that pulls into the station. The trains run at the same frequency: exactly once an hour.

You often travel between the two cities on a whim, and when you do so, you show up at the station at a completely random time. Yet after many trips over the years, you notice that you have taken one of the trains three times as often as the other. Is this just really bad/good luck, or is there another likely explanation?

#### Solution

Although the trains run at the same frequency and you show up at the station at random times, the timing of the trains’ arrival relative to each other matters.

Let’s say Train A arrives on the hour (8:00, 9:00, 10:00, etc.) and Train B arrives on the 15th minute of every hour (8:15, 9:15, 10:15, etc.).

This means Train A is the next train to arrive between :15 and :00 (45 minutes) of each hour, but Train B is the next train to arrive only between :00 and :15 (15 minutes) of each hour. Since the trains are identical and you always take the next train to arrive, you are three times as likely to arrive at the station with Train A as the next train than with Train B as the next train!

So if the schedules are set something like this, it would be natural that you consistently take one train three times as often as the other.