Cube

What is the average volume of a cube inside the unit cube if all the faces are parallel to the coordinate planes? Here are the results of an experiment with 1000\,000 trials repeated 10 times.

0.02835840.02859830.02862880.02848690.0283728
0.02855600.02842750.02853880.02837030.0289527

Let x denote the edge of a cube. A cube of this size can be moved so that the vertex closest to the point (1,1,1) can be placed anywhere inside the small cube, which has edge 1-x.

The expected volume is given by the integral

    \[\frac{\int_0^1x^3(1-x)^3\,dx}{\int_0^1\left(1-x\right)^3\,dx}=\frac{1}{35}\approx 0.0285714\]