Random inscribed triangle containing the centre

Three points are placed independently and uniformly on a circle. What is the probability that the triangle they form contains the circle's centre?

$1/4$. Fix the first two points. The centre is inside exactly when the third point lands on the arc opposite the minor arc between them, which has the same length as that minor arc. The minor arc is uniform on $[0,\pi]$ with mean $\pi/2$, so the probability is $\tfrac{\pi/2}{2\pi}=\tfrac14$.