Points within a Hypersphere

Probability Level pending

Imagine that you had 5 randomly chosen points on the surface of a 4D hypersphere. You connect these points up into a 5-cell, a 4D tetrahedron. What is the probability that the centre of the hypersphere lies within the 5-cell?

Give your answer as a decimal.


The answer is 0.0625.

William Allen by William Allen , 21, United Kingdom

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1 solution

William Allen
Dec 22, 2018

Imagine picking points as having 4 lines passing through the origin, and picking a random point which must lie within a segment. Picking either end of a line has a 1 2 \frac { 1 }{ 2 } chance and ( 1 2 ) 4 = 1 16 = 0.0625 (\frac { 1 }{ 2 })^4 = \frac { 1 }{ 16 } = 0.0625 .

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