A point is chosen at random inside a square ABCD. What is the probability that it lies inside the triangle whose base is the side AB and whose third vertex is the centre of the square?

Let the side be $s$, so the square has area $s^2$. The triangle has the whole side AB as its base and its apex at the centre, which is a distance $s/2$ from AB, so its area is $(1/2)\times s\times (s/2)=s^2/4$. The probability is $(s^2/4)/s^2=1/4$, the same for a square of any size. As a check, the four triangles built on the four sides in this way tile the square exactly, so each must take a quarter of it.