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:

    \begin{equation*} \begin{array}{ccccc} 0.028358445 & 0.028598372 & 0.028628856 & 0.028486958 & 0.028372860 \\ 0.028556010 & 0.028427579 & 0.028538886 & 0.028370322 & 0.028952778% \end{array} \end{equation*}

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

Cube

The expected volume is given by the integral

    \begin{equation*} \frac{\int_{0}^{1}x^{3}\left( 1-x\right) ^{3}\,dx}{\int_{0}^{1}\left( 1-x\right) ^{3}\,dx}=\frac{1}{35}=0.0\overline{2857\,14} \end{equation*}