Fibonacci numbers and the probability of polygon formation using random length sticks

We present two complementary proofs that, if the lengths of $n$ sticks are sampled at random, then the probability that no $p+1$ sticks can form a $(p+1)$-sided polygon can be expressed as the product of the reciprocals of a series of terms involving the $p$-step Fibonacci numbers. The first proof uses matrix algebra to extend the method previously used by Sudbury et al. to derive expressions for the probabilities of not being able to form triangles and quadrilaterals. The second alternative proof uses a different approach based on expressions for the minimum and maximum lengths of each stick that are compatible with the constraint of not being able to form a $(p+1)$-sided polygon, and provides insights into the structure of the probability expressions and the underlying reason that they include the Fibonacci numbers. Furthermore, the approach is developed in a generalised way that can, in principle, be applied to sticks randomly sampled from any probability distribution.