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

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  • #1
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
Screen Shot 2021-04-02 at 5.49.32 AM.png


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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  • #2
PeroK
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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##.
 
  • #3
mathman
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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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