Missives from a fly bottle
barang dot sg
Updated 14 July 2020
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.