Do three random fractions add up to less than one?

Question

Pick three numbers X, Y, Z independently and uniformly from [0, 1]. What is the probability that X + Y + Z < 1?

Accepted Answer

Because the three numbers are uniform on [0, 1] and independent, the point (X, Y, Z) is uniformly spread over the unit cube [0, 1]^3, which has volume 1. So the probability equals the volume of the region inside the cube where x + y + z < 1. That region is the tetrahedron (a "corner simplex") with vertices at the origin and at (1,0,0), (0,1,0), (0,0,1). Its volume is 	ext{Vol} = \frac{1}{3!} = \frac{1}{6}. (One way to see the \frac{1}{3!}: the simplex x + y + z < 1 with all coordinates positive is one of the 3! = 6 congruent pieces you'd get by ordering the coordinates, and they tile a unit-…

Do three random fractions add up to less than one?

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