Pain in the ass

Some guy posted the following problem at work:
"Given n positive real numbers, r_1, r_2, ... r_n, that sum to an
integer, what is the probability that [(sum from 1 to n) r_i] = [(sum
from 1 to n) r_i, each rounded to the nearest integer]?"

I must say that I'm making headway, but I have no idea how to phrase it
to people when I post a start on a solution. My solution involves some
pretty easy (n-1)-dimensional integrals (integrating 1 over shapes
analogous to right triangles and pyramids), and then calculating harder
(n-1)-dimensional integrals (crazier shapes) by adding and subtracting
the easy ones. I am quite far from a general solution, mind you; I've
just worked n from 1 to 6.

N = 4 is the last easily-explained case, as it involves visualizing
3-space. One can take a cube of unit volume, slice it up using 7 [2 *
(n-1) + 1] planes, and then find the total volume of some choice
sections.

With n = 5, it's easy enough to say "4-cube," but I have no idea how to
cope with the word "volume" in 4-space, nor do I know how to explain
that yes, I'm dividing up 4-space using 9 3-spaces.

For the record, here are my results, which might very well be wrong:
n Probability
1 1 (trivial)
2 1 (also trivial)
3 3/4
4 2/3
5 115/192
6 19/30
To put each beast in its native habitat, the proper denominator would
be (n-1)! * (2^(n-1)).

.

Quantcast