# Understanding the Dartboard Paradox: A Deep Dive into Probability

Written on

## Chapter 1: Introduction to the Dartboard Paradox

The dartboard paradox is a fascinating dilemma that challenges traditional concepts of probability, particularly when grappling with the ideas of zero and infinity. It raises the question: how do we calculate probabilities in a game of darts? In this article, we will outline the paradox, investigate its origins, and explore potential solutions.

This section introduces the paradox and sets the groundwork for a deeper discussion.

### Section 1.1: The Core Problem

Imagine you are throwing darts at a dartboard, and you are guaranteed to hit the board. We seek to determine the probability of landing on any specific point on it. This forms the essence of our inquiry.

As you throw darts randomly, the distribution of your hits is equally random. The chance of hitting any particular point is independent of hitting any other point. Thus, we can view each hit as an independent event.

Now that we’ve established the premise, let’s delve into the paradox itself.

#### Subsection 1.1.1: The Paradox Unveiled

In probability theory, the combined probabilities of all potential events within a set cannot exceed 1. Therefore, logically, the likelihood of hitting any single point on the dartboard must be less than one, given our assumption of random hits.

The paradox emerges when we ponder the following question:

How many points exist on the dartboard?

The straightforward answer is that there are infinitely many points. This leads to a complex consequence: summing infinite non-zero probabilities would approach infinity. Yet, we know from probability axioms that the total probability cannot surpass 1.

What if we assign a probability of zero to most points, attempting to make the sum of probabilities equal to 1? This notion leads to contradictions. First, assigning a zero probability conflicts with our initial condition of random dart hits. Additionally, if certain probabilities are set to zero, the independence of events is compromised.

This encapsulates the paradox! So, where do we go from here?

## Chapter 2: Analyzing the Roots of the Paradox

The dartboard paradox fundamentally arises from our interest in calculating discrete probabilities for an infinite set of outcomes. However, we know the total probability for these outcomes cannot exceed 1.

To explore the discrete probability of hitting any specific point on the dartboard, we encounter a fundamental issue.

### Section 2.1: Rethinking the Problem

A key observation is that a point lacks dimensionality. While a point has defined coordinates on the dartboard, it occupies no physical area. Conversely, a dart's tip covers some area.

Photo by Pixabay from Pexels

This insight prompts us to reformulate our inquiry: instead of calculating the probabilities for point hits, we should consider dart-tip-sized areas on the board. By doing so, we can establish a finite number of areas, allowing us to assign finite probabilities that collectively sum to 1.

This approach offers a solution, albeit one that requires reframing the original question. What if we wish to address the original issue directly?

### Section 2.2: A Novel Approach to the Paradox

If we consider the dartboard as an infinite space of points, we can apply the concept of infinitesimals to approach our dilemma. For a deeper understanding of infinitesimals, refer to my article: What Really Happens When You Invent Infinite Infinities?

The probability of hitting each point could be represented as an infinitesimal, allowing the sum of all probabilities to equal 1. This method seems promising!

However, introducing infinitesimals challenges the foundations of conventional probability theory, leading us to Nonstandard Probability Theory—a less traditional mathematical approach.

## Chapter 3: Final Thoughts

Ultimately, we have identified two potential methods for addressing the dartboard paradox. The first involves redefining the problem, while the second requires us to reconsider the fundamental concept of probability.

Under traditional probability frameworks, a true solution to the dartboard paradox seems elusive without altering the original inquiry. This demonstrates the limitations of our logical constructs.

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

I trust you found this exploration enlightening. If you wish to support me as an author, please consider clapping, following, or subscribing.

For further reading, you might enjoy: Is Zero Really Even Or Odd? and Can You Really Solve The Staircase Paradox?

This video titled "Making sense of the Dartboard Paradox" provides insights into the complexities of this paradox and its implications.

This video titled "Dart Board Probability" further explores the intricacies of probability in relation to the dartboard paradox.