Right triangle $XYZ$ has legs of length $XY = 12$ and $YZ = 6$. Point $D$ is chosen at random within the triangle $XYZ$. What is the probability that the area of triangle $XYD$ is at most $12$?
In order for the area of triangle XYD to be at most 12, the height of triangle XYD is at most 2. The height of triangle XYD can be 6, so the probability is 2/6 = 1/3.