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…