Talk:Buffon's needle

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Probability distribution[edit]

There's an assumption here that we can use a uniform probability distribution for each variable. For rotation that seems plausible, but for x it doesn't seem plausible at all. Luke (talk) 22:51, 29 January 2011 (UTC)

Another proof[edit]

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 , the distance between the lines. The length of the needle is then . 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 . Of course, its length is times shorter. Thus, on average it will get times less hits — its probability of being hit on one throw will be . By throwing it times, we'll get (on average) 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)
This proof can be justified with linearity of expectation. 2607:F140:400:A007:9003:4374:B218:4EE3 (talk) 18:24, 5 December 2014 (UTC)

Likelihood vs certainty[edit]

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)

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

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. 18:04, 17 August 2007 (UTC)

Even better new proof[edit]

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 (talkcontribs) 11:23, August 30, 2007 (UTC)


How is the probability distribution in x:

?--Loodog (talk) 19:34, 27 November 2007 (UTC)

On Lazzarini's "randomness" / fairness / truthfulness[edit]

I answered my own question, since nobody responded to my query: the section titled "What is estimate of departure from 'fairness' of Lazzarini's needle?" -- which I posted August-2007. It is now March-2009. I was kind of tired of getting no response no matter where I went for one. And I WAS truly curious about my question. So, lacking any response, I tackled it myself.


OK, you have a coin -- a biased coin. It will give a head with probability 0.53051647697298445256294587790838 ( which is 5÷(3·pi) ).

You agree that you will flip this coin in SETS of 213 tosses apiece.

You will stop when BOTH these two conditions are satisfied (forget about WHY; although you folks know why):

(a) an entire "set" of 213 tosses is complete; and

(b) the SUM of all the heads of all the tosses -- from the current AND ALL prior sets -- is an EXACT WHOLE NUMBER MULTIPLE of 113.

What is the probability of you stopping after 1 set, after 2 sets, after 3 sets,...?

What is the probability of you stopping in 2 sets or less, 3 sets or less, 4 sets or less...?

If that Lazzarini guy took 16 sets before stopping, what is the probability he was perfectly random -- or a more-likely-to-be-pertinent question (given certain other facts about his "arbitrary" setup and secret-to-himself stopping rule): what is the probability Lazzarini told the truth that the sum after the 16th set really did end in success by equalling 1,808 "heads" in 3,408 tosses (satisfying secret stopping rule (b))?


The probability that Lazzarini would have stopped in 16 trials or less if he were (a) truly random (b) more importantly, "on the up-and-up" (c) i.e., truthful! was 0.2884 -- or 28.84% chance Lazzarini would have met his goal by 16 trials or fewer or 71.16% chance he would NOT have met his goal by the time 16 binomial trials of 213 tosses had elapsed. Given how Lazzarini cheated on his setup and on his tacit secret stopping rule, are we really to take Lazzarini's word that he got EXACTLY 1,808 needle crossings on his sixteenth trial (of course he didn't say "my 16th trial"; he said "after 3,408 tosses")? The chances that Lazzarini would have achieved his goal was 28.8% by 3,408 tosses or fewer (in sets of 213). Or odds are roughly 2½:1 (two-and-a-half to one) that Lazzarini cheated (lied) also in his ending as well as in his cheated setup and secret stopping rule. That is 2½:1 Lazzarini said something like this to himself "I'm shy by just two needle crossings; let's just say they crossed instead of going on to yet another (17th!) trial of 213 throws; what's two needles? call them 'crossed' and be done with it; otherwise I'll never be able to quit." Of course, there is 28.8% chance Lazzarini did not lie/cheat at the end. But like I say, given what he did at the beginning and the odds being 2.4 to 1 against Lazzarini achieving success, we have good cause to expect non-truthfulness at the end.

The basis of my probability was a simulation of Lazzarini experiments (via Turbo Pascal 5.5 MS-DOS original program I wrote). The random number generator used was the Wichmann & Hill 1982 RNG which I have found to be superior to all other RNG's including even the Wichmann & Hill RNG of 2005! (Incidentally, fairly recent Excel versions have used the WH1982 RNG.) For any reader who has uncovered an earlier edit of this section, you would see that I said there that Lazzarini's success in 16 or fewer binomial trials was "definitely more than 0.09, but most likely less than 0.32 -- but how much less is not known". This claim was based on analytical computations rather than simulation. Unfortunately, analytical computation was EXACT only for Prob{stop @ n=1} and for Prob{stop @ n=2} only. For stopping probability for all higher values of n (n of 3 or more) I made an approximation, but for which I did not know the error. That the cummulative probability was greater than 1 by n of about 130 or so suggested that the probabilities were overestimated or that the probability estimates did not decrease fast enough with increasing n in my computations. This excess probability (greater than 1.0) is why I said "most likely less than" when saying "0.32". I used MathCAD 6.0e for the computations.

Finally, for the two cases wherein I KNOW the exact probability (n=1 and n=2), here is the comparison of those exact values and the simulation I did:

n=1 EXACT Prob{stop @ n=1} = 0.054708 Simulation: 3,559 / 65,534 = 0.054308 (for exact, simulation should have had 3,585 occurrences)

n=2 EXACT Prob{stop @ n=2} = 0.035714 Simulation: 2,312 / 65,534 = 0.035279 (for exact, simulation should have had 2,340 occurrences)

I can give more details on my simulations and computations. Just ask. (talk) 17:01, 4 March 2009 (UTC) (talk) 00:00, 3 March 2009 (UTC)

I think that you are possibly not catching the point of Lazzarini's estimate. As far as I understand (I admit I do not have enough reference on this), Lazzarini's point was to show how a delicate matter is a statistical experiment; it is not even relevant if he really performed the needles' count or that was meant just as a thought experiment. In particular, notice that the preceding experiments with Bouffon's needles had made no arithmetic consideration at all on the effect of the choice of the lengths: which is on the contrary quite a relevant point, as we know after Lazzerini's remark. Notice also that the astonishing precision of Pi he got (or could have got) was so in contrast with the other ones (2-3 correct digits at most, with many more needles), that it could by no means pass unremarked. Did he want to be reputed a kind of magician by the international scientific community, is this your idea?
Let me add that, although in the current years of deep decadence and degeneration of Italy it is quite legitimate to doubt about any statement from an Italian source (especially if coming from the Italian government) we are speaking of the eve of 1900, a golden period when Italian mathematicians where among the most quoted in the world, together with German and French. The idea of a scientist cheating in such a stupid manner is somehow too strange to me. I could be proven totally wrong, of course. --pma (talk) 13:22, 7 May 2009 (UTC)

From the Lazzarini Questions (& Results) Contributor:

First, I am glad to see that finally somebody besides myself has taken some interest in the Lazzarini issues. As the Article itself has pointed out, the remarkable result for the Pi estimate was rigged based on getting a particular quotient or ratio. To get this quotient, certain size needles and wood strips were required. Since the quotient could be physically fairly easily obtained since the numerator and denominator were both less than 300, then if one got exactly the right under-300 "numerator" and exactly the right under-300 "denominator" one would get an amazing value for the Pi estimate.

But none of the above was MY issue. MY issue was HOW GOOD A RANDOM NUMBER GENERATOR WAS LAZZARINI? There are three overall kinds of random number generators: physical devices operated by humans; physical processes that nature does whether or not a human is even around; algorithms. In computers, random number generators (RNG's) are the last type: algorithms. Whether they be linear congruential, "multi-cycle" congruential (like Wichmann & Hill) or something complex like the highly praised Mersenne Twister (JMP Statistical software uses the Mersenne Twister for its random rows selection [JMP Technical Support told me this!]; at least one fairly recent Excel version uses Wichmann & Hill's 1982 RNG [Excel for Chemists book]) -- these are all algorithms. It is perfectly fine to ask "how good is the RNG?" There are a plethora of "goodness" criteria for these RNG's. RNG testing is typically done with a "battery" of tests. Hence an older, but venerable, testing suite was called "DIEHARD". While the batteries of tests could test ANY RNG, you need to get the RNG values into a computer. Algorithmic RNG's obviously very easily put the RNG values into the computer. (If you want you can spin your spinner and type all the values you got into the computer and then run DIEHARD statistical tests -- but that would be tedious, to say the least.) Quantum decays or radioactivity are natural processes that don't involve a human and are generally considered truly random and would be tested only to see the "power" of the tests -- i.e. you'd be testing the tests if you applied them to "perfectly quantum mechanically random physical generators". But last comes a physical device whose definition IN CONCEPT is perfectly random. But as a real physical device -- IS IT truly as random as the concept?. Is your Twister (the old party game with hand/foot color dots) transformed by markings to a 0 to 1 scale truly random? Are there some parts of the spinner that are more likely than others to stop the indicator? Does the way the person spinning the indicator spin it deviate from perfect randomness? These are all legitimate questions. The only reason some ad hominem crept into my analysis is that it was a HUMAN doing a physical thing. One can question the merits of the physical thing from ideal perfection, but, so too, one can question the human's technique -- did the technique impose a non-randomness? From my results, for Lazzarini to accomplish his goal (the exact quotient he required) in 16 or fewer sets was a probability of 28.8%. The probability is not miniscule, but it is certainly well under 50%. Did Lazzarini get lucky? The probability is what it is. But given the riggining of why he did the experiment in sets of 213 tosses and so on (to get an "exact" pi) it is likely that he got tired of doing yet another set of 213 tosses and probably chose to call cross or not cross a couple/few needles so he could quit. Had my computation been that the probability was, say, 82% of getting the requisite ratio in 16 or fewer sets (Lazzarini required 16 sets) then I would have said this: "it is highly likely that Lazzarini and his tossings were random". Actually, since this is a stopping rule problem, if -- just say, becasue it isn't like this -- probability of success was 90% by only THREE sets yet Lazzarini needed 16 sets to achieve the goal, then Lazzarini would NOT have been random (but in the other direction -- taking too many sets of 213 tossings). In this other direction (doing the experiment too long) before success (IF IT WERE THE CASE THAT FEWER SETS WERE PROBABILISTICALLY REQUIRED, BUT THAT IS NOT TRUE HERE), then I would say "Lazzarini wasn't random". But I would not say he cheated because who would cheat in the sense of saying "oh, I have my success and got the quotient goal, but I'll say I didn't and do oodles more sets of 213 tossings". But, as it is, success is to be had with 16 or fewer sets of 213 tossings, only with a probability of 28.8%. Do you think Lazzarini really got that success after set #16 or is it more likely that he had had enough and called one or two needles "cross" or "no cross" as needed to finally be able to quit? The fact that he is Italian is something I do not even consider. Considerations are (a) the probability of achieving success by 16 or fewer sets of 213 physical needle tossings (b) with a low probability of success a human gets tired of keeping going and (c) Lazzarini's "history" of having rigged his "experiment" such that the quotient would give an excessively precise value of Pi doesn't speak well of "giving him the benefit of the doubt" that when he stopped he did so legitamately when he had only a 28.8% probability that he should have stopped by the point he did (or sooner). (talk) 20:06, 8 June 2009 (UTC)

Earlier attempt to compute pi using Buffon's Needle

On June 5, 1872 Asaph Hall, the astronomer, submitted an article entitled "On an Experimental Determination of Pi" to the journal Messenger of Mathematics. The article appeared in the 1873 edition of the journal, volume 2, pages 113-114.

This is described in the wikipedia article on Asaph Hall —Preceding unsigned comment added by Frederika41 (talkcontribs) 20:36, 8 December 2009 (UTC)

On June 5, 1872 Hall submitted an article entitled "On an Experimental Determination of Pi" to the journal Messenger of Mathematics. The article appeared in the 1873 edition of the journal, volume 2, pages 113-114. Frederika41 (talk) 20:40, 8 December 2009 (UTC)

Early attempt at using Buffon's Needle[edit]

On June 5, 1872 Asaph Hall (the astronomer) submitted an article entitled "On an Experimental Determination of Pi" to the journal Messenger of Mathematics. The article appeared in the 1873 edition of the journal, volume 2, pages 113-114.

This is described in the Wikipedia article on Asaph Hall.

(I apologize for my earlier clumsy attempts to post this comment - my first and probably last time to post a comment). Frederika41 (talk) 20:47, 8 December 2009 (UTC)

I think there's a mistake in the "using elementary calculus" section[edit]

Since the integral is with respect to x, you need the pdf of x over (0, L/2), not the integrated probability. For the special case t=L,that pdf is 2/L, so that the overall probability of a cross is 2/L times the integral. The integral is L/pi, so the overall probability is (2/L)*(L/pi)=2/pi. — Preceding unsigned comment added by (talk) 01:02, 21 February 2012 (UTC)

Buffon quote[edit]

Where is the English Buffon quote from? It is uncited, and seems to bear no relationship to the French original. -- (talk) 04:52, 13 September 2016 (UTC)

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