Two Girlfriends and Two Trains

Unfaithful Bob has two girlfriends Alice and Carol. Bob can take a train ride from the city’s train station to get to them. There is only 1 line going in both directions: the eastward line will get him to Alice’s place, and the westward line will get him to Carol’s place.

Each weekend, Bob will show up at the train station at a random time and take the first train that arrives. He figures since both trains arrive at a regular interval of 10 minutes, on average he’d be visiting them both equally.

However, after a year, Carol complains that Bob only visited her 12 times that year: much less than half of the weekends! How is this possible?

Hint

While this scenario could occur because of pure chance, there is another probable cause for this abnormality.

Solution

There are many reasons that this could be happening, but we’ll just examine a simple scenario just to show that it’s possible. Let’s consider the following schedule for 9 AM in the morning:

Westward train: 9:00, 9:10, 9:20, 9:30, 9:40, 9:50
Eastward train: 9:01, 9:11, 9:21, 9:31, 9:41, 9:51

If Bob arrives at any time that ends with a 0 (in the minutes place), he will end up taking the westward train. Otherwise, if the minute ends with a 1 to 9, he ends up taking the eastward train. This means with a schedule like this, he will, on average, end up taking the eastward train 9 times more often than the westward train.

(Notice that Bob will only take the westward train if he arrives in the shaded area, and the eastward train otherwise).

In fact, if one train always comes almost immediately after the other, then Bob will almost never take the earlier train. Oh well.

Two Girlfriends and Two Trains

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