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.)

The probability is (1/n!), at least for n=1, 2, 3.

For n=1, it's obvious that the sum will never exceed 1.

If n=2, we can call the two random numbers x and y, and they determine a point within a unit square. If the point is below the diagonal with equation x+y=1 that goes from (0,1) to (1,0), the sum is less than 1, so the answer in this case is (area of the square)/(area of the triangle below the diagonal) = 1/2.

For n=3, calling the three points x, y and z, they mark a point within a unit cube. If the point is below the plane with equation x+y+z=1 that passes through vertices (0,0,1), (0,1,0) and (1,0,0) the sum will be less than 1, and the answer is (area of the cube)/(area of the pyramid below the plane) = 1/6.

Posted by e.g. on 2004-07-29 11:17:52