Start with two white balls and one black in a bag. We take one ball at a time, and we finish if it is white, otherwise we replace it along with another black and another white ones (note that in every step we have two more balls in the bag than in the previous one) and keep on taking balls.
The expected value of black balls in the bag when we finish is $\pi/2$. The probability of taking a last white ball different from the initial two is $\pi - 3$, hence the probability of the complement is $4-\pi$. All of this is rather easy to show by using that $\sum_{n\ge 0}\frac{n!}{(2n+1)!!} = \frac \pi 2\,$ (in a way it is a probabilistic interpretation of the sum).
My question really is for references on this in case it was known. It is certainly easier to perform as compared to Buffon's needle, even more accurate (the variance is smaller, since $(\pi-3)(4-\pi)<(1/\pi\,(1-1/\pi)$).
Furthermore, it is even simpler to define a similar variable with expectation $e$: start with one white and one black balls and finish when we get white, otherwise replace the black alongside another white and take a ball again. The distribution of the number of balls we use is as in the case of Gnedenko's variable (number of steps with length in $(0,1)$ --independently and uniformly distributed-- we need to complete length 1).
Any reference would be welcome. I wrote down most of this as a note, but I find it too elementary to have been unnoticed before.
