What's the probability that
n random numbers from [0,1] will sum less than 1?
(For purists: "uniformly distributed, independent" random numbers are assumed.)
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Submitted by Federico Kereki
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Rating: 3.8333 (6 votes)
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Solution:
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(Hide)
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Picking n numbers in [0,1] is the same as picking a random point within a n-dimensional hypercube. If we want the sum to be less than 1, the point must also be within a "hypersimplex" -- a solid with a vertex at the origin and the other vertices at (1,0,0...0), (0,1,0,...0), (0,0,1,...0), up to (0,0,0,...0,1).
(This solid is obtained intersecting the hypercube with the hyperplane given by x1+x2+x3+...+xn=1.)
You can find the volume of this solid using simple integration, and it works out to 1/N!.
This
pyramid has 1/(N!) the volume of the cube, so the probability is 1/(N!). |
