The ngram CC classifier performed similarly to manually developed CC classifiers and has advantages of rapid automated creation and updating, and may be used independent of language or dialect.
In this paper we prove that the point in a triangle T maximizing the probability that it lies within a randomly chosen triangle within T is the center of T. A simple general closed expression for the indicated probability at any point P of T is obtained, though the proof proceeds by a symmetry argument on contributions to derivatives of the probability. It is also shown that the indicated probability is linearly related to the average over points in T of the product of the areas within T on either side of the line determined by P and
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This gives the proof for n = 3 of the following conjecture of A. Prékopa:
Fix a point P in an n − 1 dimensional simplex S. Take n independent points, uniformly distributed in S, and consider the probability ρ(P) of P being inside the random simplex determined by these n points.
Conjecture: ρ(P) is maximal, when P is the center of gravity of S.
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