Three points are chosen at random inside a square. Each point is chosen by choosing a random x-coordinate and a random y-coordinate.

A triangle is drawn with the three random points as the vertices. What is the probability that the center of the square is inside the triangle?

A quick guess would suggest that no triangle within a square can include more than half the area of that square, so (perhaps) on average the included area would be a quarter of the square, so (perhaps) the odds of a given point being within a square are 1:4. How random is random?