Choose a point uniformly at random from the unit square 0101

Choose a point uniformly at random from the unit square [0,1]×[0,1]. What is the probability that it lies inside the circle with center (0.5,0.5) and radius 0.5?

Solution

This same argument works for any region E of the unit square.

For example, suppose E is the circle with center (1/2, 1/2) and radius 1/2. Then the probability that our random point (x, y) lies inside the circle is equal to th!e area of the circle, that is,

P(E) = (1/2)^2 = /4 . If we did not know the value of , we could estimate the value by performing this experiment a large number of times