How To Solve The Dartboard Paradox?

The dartboard paradox is one of those pesky paradoxes that revolves around the concepts of zero and infinity. What makes this paradox extra special is that it involves the notion of probability as well. That’s right. What is a game of darts without calculating probabilities?

In this article, I will first brief you on the conundrum we face with this paradox. Then, I will proceed to analyse what causes this paradox. Finally, we will look at ways of solving or eliminating the dartboard paradox.

The Problem Statement

Imagine that you are to randomly throw darts at a dartboard. You are 100% certain to hit the dartboard. We are now interested in calculating the probability of your hitting any point on the dartboard. This is pretty much the problem outset.

Since you will be throwing darts randomly at the dartboard, the distribution of the hits on the board can be considered random as well. The event of your hitting any particular point on the dartboard does not affect the event of your hitting any other point on the dartboard. Therefore, we could consider the event of hitting any given point on the dartboard an independent event.

Now that the stage is set, let’s look at what the paradox is about.

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The Dartboard Paradox

In probability theory, the sum of the probabilities of all possible events in a set cannot exceed ‘1’. Therefore, logically speaking, the probability of hitting any particular point on the dartboard has to be lesser than one (under the assumed random dart-hit distribution).

The trouble arises when we ask the following question:

How many points are there on the dartboard?

The answer is simple: there are an infinite number of points on the dartboard. The consequence is not so simple: the sum of infinite non-zero probabilities would tend towards infinity as well. But we know from the axioms of probability theory that the sum of the probabilities of all possible events for the dartboard cannot be greater than ‘1’.

What if we assume a probability of zero for most of the points and somehow derive the sum of all independent probabilities to be ‘1’? Well, this would lead to a few contradictions. Firstly, zero probability for specific points goes against our initial condition of a random dart-hit distribution. And by setting some specific probabilities to zero, we could argue that the individual events are no longer independent of each other.

This is essentially the paradox! So, what now?

What Caused the Dartboard Paradox?

The dartboard paradox occurs essentially because we are interested in computing discrete probabilities of the members of an infinite set. But on the other hand, we know that the sum of all probabilities of the members of this set cannot exceed ‘1’.

If we are interested in the discrete probability of hitting any given point on the dartboard, there is simply no other apparent way to proceed.

It is, therefore, reasonable to question our original motivation and initial conditions. Let’s do that.

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A Potential Solution

The first thing to note is that a point is a dimensionless quantity. Sure, any point has its two-dimensional location coordinates on the dartboard, but it does not have any area associated with it. On the other hand, the tip of the typical dart occupies an area and not just a point.

Using this realization, we could reframe our original question to compute the discrete probabilities of dart-tip-sized areas on the dartboard instead of points. If we do this, we can be fairly certain that there will be a finite number of such areas on the dartboard. Consequently, we would be able to assign a finite probability for each area such that the total sum of all possible probabilities leads to ‘1’.

Well, this IS a solution, but it requires us to cheat a little bit by reframing our original question. What if we wanted to somehow push on with the original question?

The No-Cheating Solution to the Dartboard Paradox

Considering the dartboard as an infinite point-space, we could employ the concept of infinitesimals to solve our problem. If you are interested in learning more about infinitesimals, I’ve written more about it in this article: What Really Happens When You Invent Infinite Infinities?

The probability of hitting each point in the dartboard would be represented by an infinitesimal number such that the sum of all such probabilities would add up to 1. This looks promising!

But the challenge here is that as soon as we employ infinitesimals, the axioms of conventional probability theory break down. We are then forced to turn to an unconventional branch of mathematics known as Nonstandard Probability Theory (more information under the references at the end of the article).

Such branches are nothing new to mathematics. If it interests you, I have covered how Nonstandard Analysis vindicated the long-banned concept of infinitesimals in the mathematical world in the thrilling story of calculus.

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Final Remarks

In the end, it appears that we have two potential ways in which we can solve/eliminate the dartboard paradox. But note that the first method involves redefining our problem statement. Furthermore, if we let the discrete areas shrink down to an infinitesimal size, we end up with the original problem we had with the points.

On the other hand, our second potential solution involves redefining the very notion of probability. So, I’d say that we have a shallow victory here if we can consider it a victory at all.

Strictly speaking, under the conventional notion of probability, there is no real solution to the dartboard paradox without altering our initial question. This just goes to show how limited we are in terms of our logical abstractions.

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References: Brian Skyrms (research paper) and Benci et al. (research paper).

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Further reading that might interest you: Is Zero Really Even Or Odd? and Can You Really Solve The Staircase Paradox?