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.
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Excellent post, thank you for sharing! Some thoughts:
1) actually, as Evi explained to me, dart boards consist of fibre bound together. The darts stick in between the fibres, so actually with this metaphor, there exist a finite number of potential points where the darts can end up 🙂
2) The paragraph starting with “what we can do” isn’t entirely clear – why would we “do” that? You just countered a critical comment, but it’s not entirely clear to me where the narrative is going from here.
3) I personally find the generalisation from this specific example, where you show how resolving this issue requires subjective consensus, to ‘all probability’, dubious. I think it’s quite possible that probability is objective, yet the example you provided happens to have some properties rendering probability /in that example/ subjective. So I’m not entirely convinced yet that all of probability is subjective 🙂
Thanks for your comments!
Let me begin by (meta-)commenting on your point 2) (“why ‘do’ the dividing?”). Hopefully my answer will also address your point 3) and then we will be in a good position to address point 1) (perhaps you meant this in a tongue-in-cheek fashion, but it is a very good point).
You’re right, I could have been clearer in the 2nd paragraph. The reason for ‘doing’ it (making the division in the first place) is that the possibility of probabilistic reasoning depends on it. Traditionally the degree of probability is defined as the number of favourable cases divided by the number of possible cases. For example, the probability of throwing an even number of eyes with a fair die is ½ Because there are six possible outcomes of which three are even numbers.
In every other situation in which we want to reason about probabilities we must also determine how many possibilities there are and how many of those are favourable. That is why we must divide the area of the dartboard. Firstly we must determine how many points there are which can potentially be hit by a dart: this gives us the number of possible cases. The number of favourable cases of course depends on the question: if we are interested in the result of only one throw this number equals one.
I hope that the above shows why it is justified to generalise our results to all situations in which traditional probability is applied: in all these situations we need the two conventions on which probabilistic reasoning is founded. We need to determine the number of cases (and this often involves subdividing infinite sets) and we must assume that the different cases are equally probable.
And now for point 3), which is perhaps the most important of all. Your friend states that the dartboard consists of a finite number of smaller areas because it is made out of discrete fibres. Let us assume that one of the dartplayers is a physicist and let us ask her whether she can be sure that the fibres are separate. Where does one fibre stop and where does the area in between the fibres begin? The problem is that we can never be sure because of the finite resolution of our measurements (we can’t even be sure that discreteness exists at all; perhaps what we conceive of as material particles are actually excitations in some universal matter-field). And although these botherations may sound like nitpicking in the case of the dartboard, for a physicist working with nuclear particles it is not so obvious where the transition lies between matter and space (just think of electron-clouds; where is an electron at any given point in time?). So whereas it may be reasonable to approach the dartboard with the help of traditional probability, the scientist cannot do so.
One last remark: you suggest that I am trying to convince you that probability is a subjective affair, but that is not the case. What I’ve been trying to show is that the application of probability theory depends on certain conventions and that these conventions are subjective because they can be differently chosen by different individuals. However, that does not mean that probability in toto is a subjective affair: given certain conventions a certain degree of probability is an objective fact.