2003 AMC 12A Problem 16

The triangles $\triangle ABP,$ $\triangle ACP,$ and $\triangle BCP$ have equal bases (the sides of the equilateral triangle), so their areas are proportional to the distances from $P$ to those sides. By the threefold symmetry of the equilateral triangle, $\triangle ABP$ is the largest with the same probability as each of the other two, so that probability is $\dfrac13.$ Thus, the correct answer is C.