Pick randomly three points on the unit sphere. The geodesic triangle with these points as vertices divides the sphere into two parts. What is the probability that the north pole is contained in the smaller of these parts?

For instance in the picture the red triangle does not contain the north pole.


Level 3/5

Note: The same question can be raised for an arbitrary convex subset of the plane instead of the sphere. In this case the triangles are normal Euclidean triangles and the “north pole” is a fixed point in the set. In this case the calculation of the probability is much harder. You may consider this as a hint.

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