Talk:Buffon's needle

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[edit] Another proof

There's a more intuitive way to prove that such a method works. I found it in Gerhard Niese's book 100 Eier des Kolumbus (in Estonian: 100 kolumbuse muna, Tallinn, Valgus, 1985). It's based on the fact that every part of the needle has an equal probability to cross a line. And this probability does not change if we bend the needle! Every millimetre of the needle would still get the same amount of "hits". And it's also intuitively clear (for me at least) that as we make the needle, let's say, k times longer, the number of hits is also approximately multiplied by the same k. And we may still bend it as we like. So, let's imagine a circular "needle" with a diametre of exatly t, the distance between the lines. The length of the needle is then πt. It's clear that with every throw it gets exatly 2 hits (it hits either one line 2 times or two neighbouring lines). And now let's consider another needle, this time a straight one with length \ell. Of course, its length is \frac{\pi t}{\ell} times shorter. Thus, on average it will get \frac{\pi t}{\ell} times less hits — its probability of being hit on one throw will be \frac{2}{\frac{\pi t}{\ell}} = \frac{2 \ell}{\pi t}. By throwing it n times, we'll get (on average) \frac{2 \ell n}{\pi t} hits, the same number as in the article. What do you think about this proof? Are there any holes in the logic? — Pt (T) 19:16, 27 December 2005 (UTC)

It looks good to me, Pt. Although I read it a very long time ago (when I was about 10 years old), I remember seeing the exact same argument in one of Martin Gardner's little books. That one was written in English, and probably published about 1955 or so. When I scare up the title of the book I'll post it here. DavidCBryant 22:42, 26 November 2006 (UTC)
This proof also appears in Proofs from the Book, by Aigler and Ziegler, ISBN 3540636986. -- Dominus 22:49, 26 November 2006 (UTC)

[edit] Likelihood vs certainty

From the article:

This is an impressive result, but is something of a cheat. ... Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate".

The first sentence implies that Lazzarini was definitely cheating. The second one says it merely seems likely.

Is this an internal inconsistency? DavidCBryant 23:01, 26 November 2006 (UTC)

I didn't intend it as one when I wrote that last year. What I meant was that the result itself is deceptively accurate, because it is correct to six places, because of a fluke in the numbers, when normally you'd expect to have to do millions of trials to achieve such accuracy. But whether Lazzarini deliberately adjusted the numbers to achieve such a deceptively accurate result, we don't know. If you can think of a clearer way to phrase this, please go ahead and change it. -- Dominus 00:16, 27 November 2006 (UTC)
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