Probability that centre of the square lies inside the circle joining the two points inside the square Two points are uniformly and independently distributed (located) inside a square. A circle is drawn such that the segment joining the two points is a diameter. Find the probability that the center of the square lies inside that circle.

Although the answer has already been given, it's possible to check this via simulation.  In R:
n = 10^8
x1 = runif(n, 0, 1)
x2 = runif(n, 0, 1)
y1 = runif(n, 0, 1)
y2 = runif(n, 0, 1)
x.center = (x1+x2)/2
y.center = (y1+y2)/2
radius = sqrt((x.center-x1)^2+(y.center-y1)^2)
distance.to.center = sqrt((x.center-1/2)^2+(y.center-1/2)^2)
sum(radius > distance.to.center)

This simulates drawing $10^8$ pairs of points (x1, y1), (x2, y2), and finds the center and radius of the corresponding circles, and checks to see whether those circles contain the center of the square (1/2, 1/2).  When I ran this I got 49995062, meaning that the probability can be estimated as 0.49995062.  I'd say that's close to 1/2.
