Steve Gray writes: << I don't see that the events [A lying in BCD, B lying in ACD, C lying in ABD and D lying in ABD, where A,B,C,D are (randomly chosen) points in a square] are disjoint.

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Sure they are, except for a set of measure 0 which we'll ignore. If e.g. A is in the interior of triangle BCD, then e.g. D cannot be in the interior of the triangle ABC. Etc. --Dan