Right, I was taking it as given that the problem of choosing a hemisphere canonically for a point meant "such that the argument works in the same way as for the circle".

Bertrand paradox just doesn't apply here, there's a natural measure on the circle and all higher dimensional spheres. I wouldn't expect an article on this subject to need to make that clarification unless it's dealing with chords or some other situation without a natural measure.

If I choose my four points as the endpoints of two chords chosen by the "random radial point" method described on the wikipedia page, is it still true that the odds of all four being covered by a semicircle are 50%?