Random point in a cylinder lying in its cone

A point is chosen uniformly at random inside a solid upright cylinder of radius $r$ and height $h$ whose bottom face is centred at the origin. Inside the cylinder sits a cone with the same base and the same height. What is the probability the point lies inside the cone?

$\tfrac13$. For a uniform point the probability is a volume ratio, and a cone has one third the volume of the cylinder with the same base and height: $\tfrac13\pi r^2h/(\pi r^2h)$.