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A run is a sequence of more than one consecutive identical outcomes, also known as a clump. Let R_p(r,n) be the probability that a run of r or more consecutive heads appears ...

Bayesian analysis is a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a "prior ...

The Buffon-Laplace needle problem asks to find the probability P(l,a,b) that a needle of length l will land on at least one line, given a floor with a grid of equally spaced ...

A formula for the Bell polynomial and Bell numbers. The general formula states that B_n(x)=e^(-x)sum_(k=0)^infty(k^n)/(k!)x^k, (1) where B_n(x) is a Bell polynomial (Roman ...

The Gaussian integral, also called the probability integral and closely related to the erf function, is the integral of the one-dimensional Gaussian function over ...

A semiperfect magic cube, sometimes also called an "Andrews cube" (Gardner 1976; Gardner 1988, p. 219) is a magic cube for which the cross section diagonals do not sum to the ...

Taking a connected graph or network with a high graph diameter and adding a very small number of edges randomly, the diameter tends to drop drastically. This is known as the ...

A formula for the generalized Catalan number _pd_(qi). The general formula is (n-q; k-1)=sum_(i=1)^k_pd_(qi)(n-pi; k-i), where (n; k) is a binomial coefficient, although ...

sum_(k=-infty)^infty(a; m-k)(b; n-k)(a+b+k; k)=(a+n; m)(b+m; n).

The Christmas stocking theorem, also known as the hockey stick theorem, states that the sum of a diagonal string of numbers in Pascal's triangle starting at the nth entry ...