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

AI Thread Summary
The discussion explores the concept of probability using concentric circles to illustrate how different coordinate systems can yield varying probabilities for the same scenario. When selecting a point randomly in a larger circle, the probability of it falling within a smaller circle can be calculated as 1/4 using Cartesian coordinates. However, when polar coordinates are applied, the probability changes to 1/2 due to the uniform distribution in angle and the squared radial distance. This highlights the importance of the chosen coordinate system in probability calculations. The findings emphasize that the interpretation of probability can vary significantly based on the method used for selection.
Frabjous
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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.

Screen Shot 2021-04-02 at 5.48.46 AM.png
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Here's the explanation.

Screen Shot 2021-04-02 at 5.49.48 AM.png

The first solution is a fraction (damn scanner!)

Oops! From Kendall Geometrical Probability (1963)
 
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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##).
 
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