Counting pseudo-Anosov mapping classes on the 3-punctured projective plane

We prove that in the pure mapping class group of the 3-punctured projective plane equipped with the word metric induced by certain generating set, the ratio of the number of pseudo-Anosov elements to the number of all elements in a ball centered at the identity tends to one, as the radius of the ball tends to infinity. We also compute growth functions of the sets of reducible and pseudo-Anosov elements.