Interpretation of $\frac{22}{7}-\pi$

Here's one in the spirit of Boufon's needle and Jaume's comment that follows more or less directly from the Dalzell integral.

Put down a tin with square cross-section and walk far away in a random direction. Shoot at it until you've hit it eight times. We call a hit "proper" if it exits the tin through the face that's opposite the one it entered through. What's the probability that exactly half of the eight hits were proper?

Let's say the tin has diagonal length 1. Parametrize the possible orientations of the tin with an angle $\theta$ ranging from $0$ to $\pi/4$, where $0$ corresponds to looking at the tin edge-on and $\pi/4$ corresponds to looking at it face-on. Note that $\theta$ is distributed uniformly. Then the apparent width of the tin is $\cos \theta$ and the apparent width of the region you need for a proper hit is $\sin \theta$. So the probability of a proper hit is $\sin \theta \, / \cos\theta = \tan \theta$.

Hence, the probability of exactly four proper hits out of eight is

$$ \binom{8}{4} \frac1{\pi/4} \int_0^{\pi/4} (\tan \theta)^4(1-\tan\theta)^4\,d\theta \\ = \binom{8}{4} \frac1{\pi/4} \int_0^1 \frac{x^4(1-x)^4}{1+x^2}\,dx \\= \binom{8}{4} \frac1{\pi/4} \left(\frac{22}7 - \pi\right).$$

I can't think of a nice reason to expect a priori that this particular setup would give you a good rational approximation to $\pi$, but there you have it.