bone-breaking problem | stick-breaking problem

Consider a line segment of length 1, and pick a point x at random between [0, 1]. This point x divides the line into line segments of length x and 1 - x. If a set of points are thus picked at random, the resulting distribution of lengths has a uniform distribution on [0, 1]. Similarly, separating the two pieces after each break, the larger piece has uniform distribution on [1/2, 1] (with mean 3/4), and the smaller piece has uniform distribution on [0, 1/2] (with mean 1/4). The probability that the line segments resulting from cutting at two points picked at random on a unit line segment determine a triangle is given by 1/4.

cube point picking | hypercube point picking | line line picking | square point picking