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

[Frabjous]

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)

 

[PeroK]

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 probability is $1/4$. But, if we choose polar coordinates, then the probability is $1/2$.

 

[mathman]

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