# Probability fun time: Proof that 1/3=1/2=1/4

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• Frabjous
In summary, the conversation discusses a solution involving probability and the use of polar coordinates. The solution involves choosing a point at random in two concentric circles and determining the probability of it being in the smaller circle. The conversation mentions that the solution is well-known and provides different probabilities based on the choice of coordinates. The correct probability using polar coordinates is uniform in angle and in r^2, not r.
Frabjous
Gold Member
Forgive me, I am not a probability guy, so I am unsure how well known this is. I was trying to figure something out and found this. I found it cool.

Here's the explanation.

The first solution is a fraction (damn scanner!)

Oops! From Kendall Geometrical Probability (1963)

Last edited:
The basic idea is reasonably well-known. For example, if we have two concentric circles of radii ##1## and ##2## and we pick a point "at random" in the larger circle, what is the probability it is in the smaller circle?

If we choose Cartesian coordinates uniformly, then the probablity is ##1/4##. But, if we choose polar coordinates, then the probability is ##1/2##.

When you use polar coordinates, the correct was (equal area) is uniform in angle and uniform in ##r^2## (not in ##r##).

## 1. How can 1/3, 1/2, and 1/4 all equal each other?

This is known as the "probability fun time" paradox, where it appears that three fractions with different denominators are all equal. However, this is not actually the case and it is a result of a mathematical manipulation.

## 2. What is the proof that 1/3=1/2=1/4?

The proof involves multiplying each fraction by a different number and then adding them together. This results in 1=2=3, which is a contradiction. Therefore, the original assumption that 1/3=1/2=1/4 is false.

## 3. Why is it important to understand this paradox?

Understanding this paradox can help us develop critical thinking skills and recognize when something appears to be true but is actually false. It also highlights the importance of being careful when manipulating mathematical equations.

## 4. Can this paradox be applied to other fractions?

Yes, this paradox can be applied to any set of fractions that follow the same pattern. For example, 1/5=1/6=1/10.

## 5. What is the real value of 1/3, 1/2, and 1/4?

The real value of 1/3 is approximately 0.333, 1/2 is 0.5, and 1/4 is 0.25. These are all different values and not equal to each other.

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