A remark on Buffon's needle problem (Q791972)

Let \(G_ n\) be a grid in the Euclidean space \({\mathbb{R}}^ n\) with coordinates \(x_ 1,x_ 2,...,x_ n\) determined by hyperplanes parallel to the hyperplanes \(H_ i\) with the equation \(x_ i=0\) separated by a distance of 2L. Let \(A_ j\) be the event: A segment of length L which will be ''thrown'' in a random fashion into the \({\mathbb{R}}^ n\) cuts a hyperplane of \(G_ n\) parallel to the hyperplane \(H_ j\). \textit{M. I. Stoka} [Quelques considérations concernant le problème de l'aiguille de Buffon dans l'espace euclidien \(E_ n\). ibid. 38, 4-11 (1983)] determined the probability of the event \(A_ j\) and calculated the variances \(D^ 2(\hat P_ n)\) and \(D^ 2(\hat P_ 1)\) for the estimators of \(P(A_ j)\) \(\hat P_ n=(nN)^{-1} \sum^{n}_{j=1}\) (number of times \(A_ j\) occurs in N independent trials) \(\hat P_ 1=\frac{1}{M}\) (number of times \(A_ 1\) occurs in M independent trials), respectively. For the case \(n=2\) see \textit{E. F. Schuster}, Am. Math. Monthly 81, 26-29 (1974; Zbl 0298.60009). Setting both variances equal Stoka obtained \(M=\sigma(n)nN\) and conjectured \(\lim_{n\to \infty} \sigma(n)=1\). We simplify Stoka's rather complicated expression for \(\sigma\) (n) and show that his conjecture is true.