Question 388720
For the angle to be acute, point S must lie outside the semicircle (but inside the square). Suppose MN = 2. Then, the area of the whole square is 4, and the area of the semicircle is pi/2. Therefore, the area outside the semicircle but inside the square is {{{4 - pi/2}}}. This region divided by 4 is {{{(4 - pi/2)/4 = 1 - pi/8 = (8 - pi)/8}}}.


On the other hand, I'm pretty sure the probability of the angle being a right angle is zero, mainly because the set of points within the square is infinitely dense...


To Edwin: I just realized my error, that the area of the semicircle is pi/2 instead of pi, and that the point must lie outside the semicircle. I have revised my answer and it matches with yours.