Hard

Probability

Let the radius of a penny be $1$. Assume that the thickness of the penny is non-negligible so that a flipped penny can land on its side. Find the thickness of the penny that, when flipped, has a $\dfrac{1}{3}$ chance of landing on its side. The answer is in the form $\dfrac{1}{\sqrt{k}}$ for an integer $k$. Find $k$.
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$\textbf{Note:}$ There is no definite answer for this question. Take the approach of inscribing the penny in a sphere.

Notes

Hint