Fun project based on a new result "Estimating $$\pi$$ with a Coin", by Jim Propp, https://arxiv.org/abs/2602.14487, Feb 26, 2026.
In this paper, Jim Propp shows that you can estimate the value of $$\pi$$ from repeated experiments with a coin, by counting lengths of sequences of coin flips which end when the number of heads exceeds the number of tails by one. Specifically, $$E\left[\frac{H_\tau}{\tau}\right] = \frac{\pi}{4}$$, where $$\tau$$ is the length of the sequence, $$H_\tau$$ is the number of heads in the sequence, and $$E\left[\frac{H_\tau}{\tau}\right] $$ is the expected value of the ratio $$\frac{H_\tau}{\tau}$$ after an infinite number of experiments.