Is this a commonly known paradox?

Question

I would like to know if the paradox below is commonly known and has a name.

Graham Priest, in his book Logic: A Very Short Introduction, at the end of chapter 12 “Inverse Probability“, asks the reader to consider the following.

��Suppose a car leaves Brisbane at noon, travelling to a town 300km away. The car averages a constant velocity somewhere between 50km/h and 100km/h. What can we say about the probability of the time of its arrival? Well, if it is going at 100km/h it will arrive at 3 p.m.; and if it is going at 50km/h, it will arrive at 6 p.m. Hence, it will arrive between these two times. The mid-point between these times is 4.30 p.m. So by the Principle of Indifference, the car is as likely to arrive before 4.30 p.m. as after it. But now, half way between 50km/h and 100km/h is 75km/h. So again by the Principle of Indifference, the car is as likely to be travelling over 75km/h as under 75 km/h. If it is travelling at 75 km/h, it will arrive at 4 p.m. So it is as likely to arrive before 4 p.m. as after it. In particular, then, it is more likely to arrive before 4.30 p.m. than after it. (That gives it an extra half an hour.)”

Priest merely mentions that this is somehow related to the Principle of indifference and that Thomas Bayes (Inverse probability) as well as Colin Howson and Peter Urbach (Probability theory) have done some work in that general area.

However, I have been unable to find any concrete information about this problem itself.

Answer 1

I don't know if this has a name, although I agree with Derek Elkins in the comments that it's closely related to Bertrand's paradox, but it can be resolved as follows. For simplicity I'll describe a discrete version of what happens first.

Suppose you think the possible speeds of the car are $50, 60, 70, 80, 90, 100$ km/h, evenly spaced and equally likely. In particular there are as many speeds below $75$ km/h as above it. What does this imply about the possible times the trip takes? Well, they are

$$\frac{300}{50}, \frac{300}{60}, \frac{300}{70}, \frac{300}{80}, \frac{300}{90}, \frac{300}{100}$$

hours respectively. These are not evenly spaced anymore; the gaps between them are decreasing. Halfway between $\frac{300}{50} = 6$ and $\frac{300}{100} = 3$ is $4.5 = \frac{300}{66.666 \dots}$, which is not $\frac{300}{75}$.

The same thing happens if you start out assuming that the distribution of arrival times is evenly spaced: then the distribution of speeds won't be evenly spaced. It is simply not possible for both of them to simultaneously be evenly spaced (which is roughly what you would get by repeatedly applying the principle of indifference).

In the continuous version, if you have a uniform probability density over speeds in the interval $[50, 100]$ then the corresponding probability density you have over times in the interval $[3, 6]$ is not uniform and vice versa.

To the extent that I see any relationship with Bayesian probability it's in asking the question of what constitutes an appropriate prior probability distribution here if one is given no information except that the average speed is between $50$ and $100$.

---

Here is another way to describe the paradox: there is no ambiguity about what the principle of indifference says to do to a finite set of possibilities. Your prior probability distribution is that every element of the set is equally likely, and this distribution is invariant under arbitrary permutations of the set.

However, there's a lot of ambiguity about how to apply the principle of indifference to an infinite set of possibilities. If the possibilities are parameterized by a real number sitting inside an interval then the uniform probability density over that interval is not invariant under reparameterization, as we saw in this case where it matters whether you think of the parameter as being time or speed. E. T. Jaynes would say that this is an instance of trying to apply probabilistic intuitions to infinite sets when they only really apply to finite sets.