Why Do Buses Come in Threes?

We've all been there; you wait at the bus stop for ages, thinking that your bus will never turn up and then eventually three buses come along at once. But why? It turns out that maths provides a very simple explanation

First of all, a small disclaimer. It's actually far, far more likely that the buses will arrive in twos. The usual saying is great; we like to over-emphasise things when complaining and three is a very powerful number in the English language (think Goldilocks and the three Bears, three attempts to guess Rumpelstiltskin's name, the rule of three when using adjectives etc.). So in this article we will start by looking at why buses tend to arrive in twos.

Let's think about an example.

Bus 1 leaves the bus depot to set off on its route. The driver has been held up and the bus is already a few minutes late leaving the depot. It then hits some unexpected traffic and by the time the bus reaches the first stop, it is several minutes late.

As the bus is late, there are now more people at the stop than would be expected because several people have turned up after the bus' expected arrival time. Usually they would have missed this bus and caught the next one, but because of bus 1's lateness, they manage to get onto bus 1.

The more people there are at a stop, the more time the bus has to wait at the stop as the driver waits for people to get onto the bus and pay for their tickets. Therefore, the bus is now running even later than planned.

Bus 1 now leaves the first stop and drives towards the second stop. It is now running even later and so the same thing happens again at the next stop. There are more people waiting at the stop than expected and so bus 1 spends more time than usual here too. This continues on for the rest of the bus route.

Meanwhile bus 2 has left the bus depot exactly on time. When this bus reaches the first bus stop, there are now fewer passengers than expected as some of bus 2's potential passengers managed to get onto the delayed bus 1. This means that bus 2 spends less time at the first bus stop than expected as there are fewer people to get on and pay and so closes the gap between itself and bus 1.

This keeps happening at each stop. Bus 1 has run late, leaving a smaller amount of time between its arrival and bus 2's arrival, so each time bus 2 reaches a stop, there are fewer people waiting for it than would usually be expected.

After a few stops of this happening, bus 1 has been held up more and more, while bus 2 is potentially going around its route slightly quicker than planned. It doesn't take long before bus 2 catches up with bus 1 and there we have it, two buses arriving at once.

The bigger the bus' route, the more stops there are to allow buses to catch each other up. Also, if it is a route with buses that run at small time intervals, then it doesn't take much for buses to start catching each other up.

So on a large route with small planned intervals between each bus, then yes, three buses at once can happen. It is just very unlikely, while two buses at once is much more likely than you would first expect.