  A simple braid is a positive braid that can be drawn so that any two strands
cross at most once. We prove that as $n \to \infty$, the proportion of simple
braids on $n$ 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 $3$-strand braids that may be of independent interest.
