dial e for Buffon
The use of Buffon’s needle to approximate π by a (slow) Monte Carlo estimate is a well-known Monte Carlo illustration. But that a similar experiment can be used for approximating e seems less known, if judging from the 08 January riddle from The Riddler. When considering a sequence of length n exchangeable random variables, the probability of a particuliar ordering of the sequence is 1/n!. Thus, counting how many darts need be thrown on a target until the distance to the centre increases produces a random number N≥2 with pmf 1/n!-1/(n+1)! and with expectation equal to e. Which can be checked as follows
p=diff(c(0,1+which(diff(rt(1e5))>0))) sum((p>1)*((p+1)*(p+2)/2-1)+2*(p==1))
which recycles simulations by using every one as starting point (codegolfers welcome!).
An earlier post on the ‘Og essentially covered the same notion, also linking it to Forsythe’s method and to Gnedenko. (Rényi could also be involved!) Paradoxically, the extra-credit given to the case when the target is divided into equal distance tori is much less exciting…
Xi'an's Og 
January 29, 2021 at 1:05 am
Buffon is the launch point of a favorite book about one of my favorite fields, D. A. Klain, G.-C. Rota, Introduction to Geometric Probability, Cambridge, 1997. There is also Waymire’s introduction and extension to “Brownian noodles” in AMM. Ideas on extensions abound, e.g., what if the walks aren’t Brownian but Levy flights or something? And, with adequate time, I’d ask questions like suppose one did semi-structured explorations of a posterior probability surface, where the local explorations were effectively such walks, but these informed bigger steps across the surface? These might include the Wang-Landau which you, Professor Robert, kindly introduced to me.
My present interests are modest. In retirement, I’ve become an amateur bryologist with a strong quantitative inclination, resurrecting some training in Botany in college. I’m interested in estimating rates of growths of moss patches, using a adaptation of a technique from forestry, line intersect sampling. To close the circle, that’s based upon Buffon’s needle, as by De Vries, 1986 (Sampling Theory for Forest Inventory. Springer Verlag).