You most probably know the typical example of computing Pi by generating random points in a square with a circle inside and looking at the ratio of points inside/outside the circle.
In most of the examples, the side of the square equals the diameter of the circle, but now I am asked to find the optimal relative size of the circle and the square in order to make the convergence speed of the Pi estimate optimal.
I have the intuition that the optimal choice is that the area of the square is two time the area of the circle, such that the probability of each point to be inside the circle is $\frac{1}{2}$. However, I am having a hard time to prove this (if this is actually right).
Does anyone know how show it?
Thanks for your help!