The probability that a dart will hit any specific point of a dartboard is zero because there are infinitely many points on the board. And yet if you throw a dart at a dartboard you’ll always hit some point (assuming you hit the dartboard). Hitting a specific point at a dartboard is highly improbable, but not impossible.
The responses to a similar statement in a previous post were mixed. Some people were delighted by its counterintuitiveness, whereas others were skeptical – what if we assume that the tip of the dart has a size that is not a mathematical point? What if it were, say, 0.1 square mm? That wouldn’t change the probability of hitting any one point, because there would still be an infinite number of points on the board. True, the probability of hitting any one of those points would increase by a certain factor, but no matter how large this factor is, the probability of is still – zero-probability.
What we can do is divide the area of the dartboard into a (finite) number of smaller areas and assume that if two of such areas are equal in size, then the probability of hitting them is also equal. There are two fundamental problems with such an approach:
- How do we know that equal areas have equal probabilities?
- There are infinitely many ways to subdivide an area. How do we know which one is the correct one?
To grasp the second of these problems it might be helpful to realise that this problem occurs whenever we try to subdivide an infinite set. Just think of the set of positive integers (1, 2, 3, 4, 5…etc.) and try to subdivide it into equal subsets. These equal subsets may have any size we choose. For example, we can choose subsets each of which contains two elements, like this: (1, 2), (3, 4), (5, 6), (7, 8)…(etc.). But we might also opt for the smaller subsets (1), (2), (3), (4)…(etc.) or the larger subsets (1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12)…(etc.). There are infinitely many choices, because we’ll never run out of integers! The same argument goes for areas (just assign an integer to every point) or any other infinite set.
Both of the questions facing the probability theorist can only be answered by adopting a suitable convention. The scientist must assume that equal areas have equal probability of being hit and she must assume that there is a preferred way to divide up a continuous area that is shared by other scientists. This shows that statements of probability do not have universal validity. The probabilistic problem of uniquely dividing up infinite sets has become known as Bertrand’s paradox.
If probability is in its core subjective, as we’ve seen, then why does science seem to possess an objective quality? One may suspect, perhaps, that the crux lies in finding the right convention. But that is not the case, since different conventions are logically equivalent; none of them is objectively better than the others. All we can do is what scientists have done ever since the dawn of science: find out by trial and error what are the most useful conventions in different situations.
For a thrower of darts the situation is simple because the conventions have already been decided upon. All players know beforehand that equal areas have equal probabilities (without being aware that this is a convention). The other convention, about the subdivision of the surface of the dartboard, has already been decided upon by the manufacturer of the dartboard; all players tacitly agree upon this conventional subdivision of the surface of the dartboard.
For the scientist – perhaps working in the field of particle physics or in cosmology – it is not obvious which conventions are useful: perhaps there is a particle for which equal size does not imply equal probability. Or perhaps two cosmologists from opposite points on the earth’s surface study some galaxy without knowing the angle from which they observe the galaxy. These cosmologists will disagree on which areas on the distant galaxy are equal (and hence they might disagree on the exact source or intensity of incoming radiation).
Science is, in the end, a capricious affair.