Putnam - Q6 - Use your intuition

Probability Level 1

If four points are randomly selected on the surface of a sphere and then connected internally , then what is the probability percent that the center of the sphere eventuate in the tetrahedron formed by the connected points?

15 10 25 12.5

A Former Brilliant Member by A Former Brilliant Member

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2 solutions

Consider a 2-D representation of the the sphere, a circle with three points randomly selected on it.

  • If you fix any two points, then you've fixed the region where the third point have to eventuate for the condition to be satisfied.
  • You also have only half the perimeter of the circle where the two points can occur - [0-180°]
  • The average of all the possible points A(a, b) and B(c, d) fixed for the third point C(e, f) to occur in the arc of love is 1 4 \frac{1}{4} (Think about it a little bit).

The idea generalizes to other dimension too.

Now you're ready to solve it yourself. You may wanna use linear algebra for a rigorous interpretation.

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See the solution in the 3Blue1Brown channel in YouTube.

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