For all allowable values of x between 0 and some finite boundary B (let's set B=1 for simplicity), you should be able to express the odds of x being chosen as the 'i-th' ranked dimension of an n-dimensional box as

p(x,i) = (n-1 choose i-1) x^(n-i) (1-x)^(i-1)

This gives a family of distributions for values selected with ranks 1 through n

Next, assume two n-dimensional boxes named "f" and "g" with randomly chosen dimensions [f1, f2 ... fn] and [g1, g2 ... gn] (ordered by rank). From here, I would ask the following questions:

• Given the n random dimension sections for f and g, what are the odds that f1 (max dimension in f) is greater than g1? (50%)
• Assuming that f1>g1, what are the odds that f2>g2?
  • This is not 50%
• Assuming f1>g1 AND f2>g2, what are the odds that f3>g3?

...and so on. Eventually, it should be possible to construct a dependent probability that one box fits inside the other.