Buffon's needle

Parametric equations

A needle of length $\ell$ falls at random onto parallel lines spaced $d$ apart, with $\ell \le d$. The probability that the needle crosses a line is $2\ell /(\pi d)$. After many throws, the rate of crossings estimates $\pi$.

$N$ is the number of throws and $n$ is the number of crossings. This is the short-needle case only.

Interaction

• Needle length ℓ and Line spacing d: their ratio sets the probability above, and how fast the estimate settles. The experiment keeps $\ell \le d$
• Throws per frame: each frame drops that many needles, marks each crossing, and adds to the count. The curve plots the number of throws against the running estimate of $\pi$
• Reset: clear the needles and the estimate
• The lines stay equally spaced, and a needle that crosses one is highlighted

What to notice

• A crossing is a needle that meets one of the parallel lines. A miss is drawn dimmer.
• While $\ell \le d$, the probability of a crossing is $2\ell /(\pi d)$. The estimate is $2\ell N/(dn)$.
• More throws pull the curve toward $\pi$. Changing $\ell /d$ changes that probability, and how quickly the estimate settles.

Related

• Conditional probability and Bayes’ theorem
• Binomial distribution to the normal distribution

Further reading

• Buffon’s needle problem (Wikipedia)