Biswa and Universe and Randomness

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Biswa has been thinking about the universe and its randomness nowadays. He dreams about some sort of objects in his sleep. These objects have a similar property. Every point on the surface of the object has similar distance from its center. If he dreams about a N dimensional object, he places N + 1 points randomly on its surface. Now he wonders what is the probability that the shape (for 2D object the shape will be a triangle, for 3D object the shape will be a tetrahedron, and so on) created by those points will not contain the center of the object? In the figure, there are 2 possible shape configurations for a 2D object.

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