Does a random triangle on a circle contain the centre?

Question

Three points are chosen independently and uniformly at random on the circumference of a circle, and joined to form a triangle. 1. What is the probability that the triangle contains the centre of the circle? 2. Generalise: for n random points joined into a convex polygon, what is the probability the polygon contains the centre?

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1/4 for a triangle. For n points, 1 minus n divided by 2 to the (n minus 1). When does the triangle miss the centre? Exactly when all three points fit inside some semicircle. So we compute the probability that the points lie in a common semicircle and subtract from 1. The diameter trick. Choose three random diameters first, then for each diameter flip a fair coin to pick one of its two endpoints. The three chosen endpoints are uniformly random and independent, so this is a legitimate way to generate the points. The advantage is that we can now fix the diameters and just count coin-flip outcom…

Does a random triangle on a circle contain the centre?

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