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)

[edit] What is estimate of departure from "fairness" of Lazzarini's needle?

My question is analogous to that of asking for an estimate of departure from fairness of a coin. If one flips a coin 1,000 times and gets 547 heads the departure from fairness is 0.547 rather than 0.500 or 0.047. Of course the confidence interval comes in; the binomial distribution too; or the Normal approximation to the binomial.

But now on to Lazzarini. First "fairness of his needle" I realize is not physically accurate. The fairness is the convolution of the needle, the wood strips, the throwing method etc. But call it "needle fairness" for short.

Our data is, based on what the writer wrote, that Lazarinni failed to get 113 crossings on his first trial of 213 tosses; likewise he failed to get a total of 2 x 113 by the end of two 213-tossings and so on. In fact, the supposition is that on 15 trials of 213 tossings he failed to get exactly n x 113 total crossings where n is how many 213-tossings trials elapsed.

Unlike the coin flipping I realize that these 213-tossing trials are NOT independent. For example, since the result was not success (exactly 113 crossings) in the first trial, Lazzarini went on to another 213-tossings trial (i.e. he is repeating trials until success). That is ONE source of the non-independence. The OTHER MORE PROBLEMATIC source of dependence is that whatever "error" occured (difference from exact 113-multiple result) of the previous trial(s), the next trial would have to err in the exactly opposite way to end the experiment. So, for example, if after seven 213-tossings trials, Lazarinni was shy of 7 x 113 by 4 (i.e. he got 787 instead of 791 total crossings), then in trial #8 he would stop only if trial #8 produced 117 crossings.

Give all the above, and the "fact" that Lazarinni required 16 trials of 213-tossings before hitting a total that was an exact multiple of 113, can a probability wizz compute the degree of departure from perfect fairness of Lazarinni's needle tossing? Likewise, can a statistics wizz give us 90% 95% 99% confidence intervals on the departure from fairness estimate? I am interested in both the magnitude of the fairness departure and its level of statistical significance. Unlike the enginerred-to-look-statistically-great pseudorandom number generators of the computer age, Lazarinni's was a PHYSICAL experiment and MOREOVER a HUMAN physical one (not a quantum random number generator). I wouldn't be surprised to find there was some significant non-fairness.199.196.144.12 18:04, 17 August 2007 (UTC)

[edit] Even better new proof

i have a better new proof for the buffon;s needle. can i add it?

it includes finding the area of f(x) 1/2 sin(theta) and thus finding the probability of the needle hitting the line is directly related to pi —Preceding unsigned comment added by Addy-g-indahouse (talk • contribs) 11:23, August 30, 2007 (UTC)

[edit] How

How is the probability distribution in x:

\frac{2}{t}\,dx. ?--Loodog (talk) 19:34, 27 November 2007 (UTC)