A coin thick enough to land on its side

HardProbability Brainteasers~10m

Model a coin as a cylinder of radius $1$ and thickness $t$ . Idealize a fair flip as the coin's orientation being uniform over all directions (equivalently, its axis points uniformly on a sphere). What thickness $t$ makes the probability of landing on its side exactly $\tfrac13$ ?

Show hints (2)+

Inscribe the coin in a sphere of radius $R=\sqrt{1+(t/2)^2}$ ; by Archimedes, $\mathbb{P}(\text{side})=t/(2R)$ .
Set $t/(2R)=1/3$ , i.e. $2R=3t$ , and solve with $R^2=1+(t/2)^2$ .

Answer

Reveal answer →

0.7071 (± 0.001)

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Jane Street, Optiver

A coin thick enough to land on its side

Answer

Related questions