A while back, on Pi Day, I sent a post celebrating the number 𝜋 and mentioned that though 𝜋 is best known for its properties in connection with the geometry of a circle, it actually crops up all over the place in mathematics, including Statistics.
Here’s one famous example…
Consider a table covered with parallel lines like in the following figure.
For argument’s sake, let’s suppose the lines are 10 cm apart. Then take a bunch of needles – or matches, or something similar – that are 5 cm in length, drop them randomly onto the table, and count how many intersect one of the lines on the table. Let’s suppose there are N needles and m of them intersect one of the lines. It turns out that N/m will be approximately 𝜋, and that the approximation is likely to improve if we repeat the experiment with a bigger value of N.
What this means in practice is that we have a statistical way of calculating 𝜋. Just do the experiment described above, and as we get through more and more needles, so the calculation of N/m is likely to lead to a better and better approximation of 𝜋.
There are various apps and so on that replicate this experiment via computer simulation, including this one, which is pretty nice. The needles which intersect any of the lines are shown in red; the others remain blue. The ratio N/m is shown in real-time, and if you’re patient enough it should get closer to the true value of 𝜋, the longer you wait. The approximation is also shown geometrically – the ratio N/m is very close to the ratio of a circle’s circumference to its diameter.
One important point though: the longer you wait, the greater will be the tendency for the approximation N/m to improve. However, because of random variation in individual samples, it’s not guaranteed to always improve. For a while, the approximation might get a little worse, before inevitably (but perhaps slowly) starting to improve again.
In actual fact, there’s no need for the needles in this experiment to be half the distance between the lines. Suppose the ratio between the line separation and the needle length is r, then 𝜋 is approximated by
\hat{\pi} = \frac{2rN}{m}
In the simpler version above, r=1/2, which leads to the above result
\hat{\pi} = \frac{N}{m}
Now, although Buffon’s needle provides a completely foolproof statistical method of calculating 𝜋, it’s a very slow procedure. You’re likely to need very many needles to calculate 𝜋 to any reasonable level of accuracy. (You’re likely to have noticed this if you looked at the app mentioned above). And this is true of many statistical simulation procedures: the natural randomness in experimental data means that very large samples may be needed to get accurate results. Moreover, every time you repeat the experiment, you’re likely to get a different answer, at least to some level of accuracy.
Anyway… Buffon’s needle takes its name from Georges-Louis Leclerc, Comte de Buffon, a French mathematician in the 18th century who first posed the question of what the probability would be for a needle thrown at random to intersect a line. And Buffon’s needle is a pretty well-known problem in probability and Statistics.
Less well-known, and even more remarkable, is Buffon’s noodle problem. Suppose the needles in Buffon’s needle problem are allowed to be curved. So rather than needles, they are noodles(!) We drop N noodles – of possibly different shapes, but still 5 cm in length – onto the table, and count the total number of times the noodles cross a line on the table. Because of the curvature of the noodles, it’s now possible that a single noodle crosses a line more than once, so m is now the total number of line crossings, where the contribution from any one noodle might be 2 or more. Remarkably, it turns out that despite the curvature of the noodles and despite the fact that individual noodles might have multiple line crossings, the ratio N/m still provides an approximation to 𝜋 in exactly the same way it did for the needles.
This result for Buffon’s noodle follows directly from that of Buffon’s needle. You might like to try to think about why that is so. If not, you can find an explanation here.
Finally, a while back, I sent a post about Mendelian genetics. In it I discussed how Mendel used a statistical analysis of pea experiments to develop his theory of genetic inheritance. I pointed out, though, that while the theory is undoubtedly correct, Mendel’s statistical results were almost certainly too good to be true. In other words, he’d fixed his results to get the experimental results which supported his theory. Well, there’s a similar story connected to Buffon’s needle.
In 1901, an Italian mathematician, Mario Lazzarini, carried out Buffon’s needle experiment with a ratio of r=5/6. This seems like a strangely arbitrary choice. But as explained in Wikipedia, it’s a choice which enables the approximation of 355/113, which is well-known to be an extremely accurate fractional approximation for 𝜋. What’s required to get this result is that in a multiple of 213 needle throws, the same multiple of 113 needles intersect a line. In other words, 113 intersections when throwing 213 needles. Or 226 when throwing 426. And so on.
So, one explanation for Lazzarini’s remarkably accurate result is that he simply kept repeating the experiment in multiples of 213 throws until he got the answer he wanted, and then stopped. Indeed, he reported a value of N=3408, which happens to be 16 times 213. And in those 3408 throws, he reportedly got 1808 line intersections, which happens to be 16 times 113.
An alternative explanation is that Lazzarini didn’t do the experiment at all, but pretended he did with the numbers chosen as above so as to force the result to be the value that he actually wanted it to be. I know that doesn’t seem like a very Italian kind of thing to do, but there is some circumstantial evidence that supports this possibility. First, as also explained in Wikipedia:
A statistical analysis of intermediate results he reported for fewer tosses leads to a very low probability of achieving such close agreement to the expected value all through the experiment.
Second, Lazzarini reportedly described a physical machine that he used to carry out the experimental needle throwing. However, a basic study of the design of this machine shows it to be impossible from an engineering point of view.
So, like Mendel, it’s rather likely that Lazzarini invented some data from a statistical experiment just to get the answer that he was hoping to achieve. And the moral of the story? If you’re going to make evidence up to ‘prove’ your answer, build a little bit of statistical error into the answer itself, otherwise you might find statisticians in 100 years’ time proving (beyond reasonable doubt) you cheated.
