Disjoint Placement Probability of Line Segments via Induction

Abstract

We give a different proof of the following recent result: Let a finite number n of line segments, the sum Ln of whose lengths is less than one, be placed onto the real line in such a way that their centers fall randomly within the unit interval [ 0 , 1 ] . Then the probability of obtaining a mutually disjoint placement of these segments, entirely within [ 0 , 1 ] , is given by ( 1 − L n ) n . The proof presented here uses induction on the number of line segments and provides insight, at each level of the induction, into the relationship between two seemingly different methods of placement: sequential random placement versus simultaneous random placement. From a purely mathematical perspective, these methods can be seen as equivalent. However, physical constraints in performing a simultaneous random placement of actual segments (e.g., toothpicks) might a priori lead to very different outcomes.