Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by V^_ = ( integral_0^1 ... integral_0^1_︸_12 left bracketing bar V(x_i) right bracketing bar d x_1 ...d x_4 d y_1 ...d y_4 d z_1 ...d z_4)/( integral_0^1 ... integral_0^1_︸_12 d x_1 ...d x_4 d y_1 ...d y_4 d z_1 ...d z_4), where the polyhedron vertices are located at (x_i, y_i, z_i) where i = 1, ..., 4, and the (signed) volume is given by the determinant V = 1/(3!) left bracketing bar x_1 | y_1 | z_1 | 1 x_2 | y_2 | z_2 | 1 x_3 | y_3 | z_3 | 1 x_4 | y_4 | z_4 | 1 right bracketing bar .
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