Suppose we sample two numbers X and Y independently, each uniformly from the interval [0, 1]. Note that this is equivalent to sampling a point uniformly at random from the unit square [0, 1] 2 . Find the probability of the following:

(a) |X − 1 2 | ≤ 1 4 ,

(b) |X − Y| < 1 2 ,

(c) max{X,Y} < 1 2 ,

(d) X 2 + Y 2 < 1 4

Sample Solution