In 2009, Shapiro posed the following question: "What is the asymptotic proportion of Dyck paths having an even number of hills?" In this paper, we answer Shapiro's question, as well as a generalization of the question to ternary paths. We find that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity. Our strategy relies on properties of the Fine number sequence and extends certain relationships between the Catalan and Fine number generating functions.