Notes from the underground
barang dot sg
Updated 28 January 2023
31. Bertrand’s paradox

A slow train trundles steadily from Bangkok to Pattaya, 120 km away.


The train is known to move at a fixed speed, but all you know of this speed is that it lies somewhere between 10 and 20 km/h. So all you know is that the journey will take somewhere between 6 and 12 hours.

Knowing only that the train’s speed lies between 10-20 km/h, you should (presumably) judge it equally likely to lie between 10-15, as to lie between 15-20 km/h. After all, you have no reason to think that its speed lies closer to either end.

Likewise, knowing only that the journey takes 6-12 hours, you should (presumably) judge it equally likely to take 6-9 hours, as to take 9-12. Again, you have no reason to favour a journey time closer to either end.

Unfortunately, these two results are inconsistent!

Notice that if the train travels at 15 km/h, the journey will take eight hours. The first result then implies that you should judge the journey equally likely to take 6-8 hours, as to take 8-12, which contradicts the second result.

The French mathematician Joseph Bertrand discovered an entire class of paradoxes of this sort in the 19th century.