Simple braids tend toward positive entropy

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as [Formula: see text], the proportion of simple braids on [Formula: see text] strands that have positive topological entropy tends toward 100%. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple three-strand braids that may be of independent interest.