Disjoint Placement Probability of Line Segments via Geometry
Christopher Ennis, John Shier · College Mathematics Journal · 2023
We have shown that when any finite number n, of line segments with total combined length less than one, have their centers placed randomly inside the unit interval [0,1], the probability of obtaining a mutually disjoint placement of the segments within [0,1], is given by the expression (1−L)n where L=|L1|+⋯+|Ln|, and |Lk| denotes the length of the k-th segment, Lk. The result is established by a careful analysis of the geometry of the event, “all segments disjoint and contained within [0,1],” considered as a subset of the uniform probability space of n centers, each of which is in [0,1]; that is to say, the unit n-cube of Rn. This event has an interesting geometric structure consisting of n! disjoint, congruent, (up to a mirror image) polytopes within the unit n-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an n-dimensional cube with common edge length 1−L, and hence an n-volume given by the formula. In the case of n = 3 segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [Citation4]. In fact for all n > 0, the polytopes comprise a partition of the n-dimensional hypercube, and are therefore n-dimensional space filling.