Rectangular Solid

Here the problem is to find the average volume of a rectangular solid inside a unit cube if all the faces are parallel to the coordinate planes.

The average length of each edge
should be about 1/3, so the average volume should be about \left(1/3\right) ^{2}=1/27.

Results of an experiment with 1000\,000 trials repeated 10 times:

0.03711140.03709230.03700930.03697290.0371264
0.03700460.03705730.03708410.03697320.0371046

Consider a rectangular solid of dimensions x\times y\times z. The probability of picking such a rectangular solid should be proportional to the volume \left( 1-x\right) \left( 1-y\right) \left( 1-z\right) of the small rectangular solid. The average volume is

    \begin{equation*}\frac{\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}xyz\left( 1-x\right) \left(1-y\right) \left( 1-z\right) \,dx\,dy\,dz}{\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left( 1-x\right) \left( 1-y\right) \left( 1-z\right)\,dx\,dy\,dz}=\frac{\frac{1}{216}}{\frac{1}{8}}= \frac{1}{27}\end{equation*}