LAST EDITED: July 20, 2019
“The dart that said zero”
The probability that a dart will hit any specific point on 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, 0.01 square mm? Doesn’t that solve the problem? I’m afraid not. The assumption that the dart has a certain size wouldn’t change the probability of hitting any one point, because there would still be an infinite number of points on the board. Of course, the probability of hitting any one of those points would increase by a certain factor, but no matter how large this factor is, a probability of is still : the probability remains zero.
What we can do is divide the area of the dartboard into a (finite) number of smaller areas and assume that the probability of hitting a certain area scales with the size of the area. This assumption boils down to assuming that if two areas are equal in size, then the probability of hitting them is also equal. Let’s call this the assumption of equal probabilities. At first sight it might seem silly to even pay attention to this assumption – let alone give it a name. Isn’t making such an assumption what we always do? A fair die has sides of equal size, divisions on the roulette table have equal sizes, etc. etc.
Indeed that is what we do in daily life. And yet it is good to realise that 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. We usually assume that equal areas have equal probability of being hit and that there is a natural way to divide up a continuous area (such as a roulette table) that is shared by everyone who looks at the situation, but these assumptions are actually conventions. The probabilistic problem of uniquely dividing up infinite sets has become known as Bertrand’s paradox.
It might seem an obvious choice that equal areas have equal probabilities of being hit. Isn’t that what we always assume? Well.. yes, but games in which this assumption seems the most obvious (such as games of dice, darts, or roulette) are chosen as games of chance precisely because the assumption of equal probability holds here. Think of this example: suppose we know that a tree in the forest is going to fall. If it falls, it does so either (slightly) to the right or (slightly) to the left. There are two possibilities, so if we know nothing about the situation, it seems natural to assume that both possibilities have a probability of . But that’s odd! We just said we know nothing about the situation, and now we can make predictions about what will happen.
Science Is A Different Game
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. It might be the case that the size of the galaxy as it appears to the two cosmologists differs by a factor two. Now suppose both cosmologists try to estimate the probability of light being emitted from the galaxy. If they were to naively adopt the assumption that probability scales with area, they will disagree!
Science is, in the end, a capricious affair.