TITLE: Is there a meaningful example of probability of $\frac1\pi$?
QUESTION [29 upvotes]: A large portion of combinatorics cases have probabilities of $\frac1e$.
Secretary problem is one of such examples. Excluding trivial cases (a variable has a uniform distribution over $(0,\pi)$ - what is the probability for the value to be below $1$?). I can't recall any example where $\frac1\pi$ would be a solution to a probability problem.
Are there "meaningful" probability theory case with probability approaching $\frac1\pi$?

REPLY [4 votes]: There are quite a few geometric probabilities related problems, involving $\pi$. The simplest I can think of, imagine this being a darts board.

The probability of hitting the square, assuming the shot didn't miss the board, is $\frac{2}{\pi}$.
If the radius of the circle is $r$, then one side of the square is $r\sqrt{2}$ and $p=\frac{S_{square}}{S_{circle}}=\frac{2 r^2}{\pi r^2}=\frac{2}{\pi}$