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The ratio of the volume of the tetrahedron to the volume of the sphere is:
\frac{V_{\text{tetrahedron}}}{V_{\text{sphere}}} = \frac{\frac{\sqrt{2}}{12} s^3}{\frac{\pi \sqrt{6}}{8} s^3} = \frac{\sqrt{2}}{12} \cdot \frac{8}{\pi \sqrt{6}} = \frac{2\sqrt{3}}{9\pi}
Thus, the final answer is \(\boxed{\frac{2\sqrt{3}}{9\pi}}\).Question:

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