On the probability of forming polygons from a broken stick

Break a stick at random at $n-1$ points to obtain $n$ pieces. We give an explicit formula for the probability that every choice of $k$ segments from this broken stick can form a $k$-gon, generalizing similar work. The method we use can be applied to other geometric probability problems involving broken sticks, which are part of a long-standing class of recreational probability problems with several applications to real-world models.