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I attended a course on stochastic processes a few years ago. During the course the lecturer mentioned that there is a mathematical proof (with some assumptions, naturally) of non-existence of a fair coin. Now I can't recall the details and can't locate the paper.

Is there such a proof?

I vaguely remember that the idea was to prove that given that coin's sides are distinguishable (by the structure, not color) one can't make the coin fully balanced.

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In the book "lying with statistics" the author shows two other possibilities for the outcome of a coin toss. One is that it lands perpendicular to the surface, the other is that a bird comes along and takes it from the air before it lands. –  Lunasaurus Rex Nov 1 '11 at 13:42
    
AT any rate, there is good experimental evidence that coin tossing is not fair in practice. But as I recall, it had more to do with the tossing itself than in the coin (i.e., the bias has to do with which side is facing up when you toss it). –  Thierry Zell Nov 1 '11 at 13:48
    
    
A similar question was asked at CrossValidated: stats.stackexchange.com/questions/30116/… –  Joel Reyes Noche Feb 27 '13 at 12:47
    
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3 Answers

up vote 8 down vote accepted

This well-known paper seems to imply that the shape of the coin doesn't really matter: http://comptop.stanford.edu/u/preprints/heads.pdf

From the lit review:

In light of all the variations, it is natural to ask if inhomogeneity in the mass distribution of the coin can change the outcome. [Lindley, 1981] followed by [Gelman & Nolan, 2002] give informal arguments suggesting that inhomogeneity doesn’t matter for flipped coins caught in the hand. Jaynes reports that 100 flips of a jar lid showed no evidence of bias. We had coins made with lead on one side and balsa wood on the other. Again no bias showed up.

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Similarly, a paper by Gelman and Nolan, You can load a die but you can't bias a coin, American Statistician 56:308-311 suggests that in practice all coins are fair. –  Andrew T. Barker Nov 1 '11 at 14:00
    
Ah, there is the catch "...inhomogeneity doesn’t matter for flipped coins" -- they limit it to coins. I thought it was about any "coin like" flat object and with some thinking I concluded that there is no "fair buttered bread" due to Murphy's law... Thank you for the reference! –  user10891 Nov 1 '11 at 18:15
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I think a fairly good demonstration is Persi Diaconis' machine to toss a perfectly standard US quarter to a single predetermined side with something like 99% accuracy. I have heard it said that he could do it himself by hand years ago when he had practiced extensively. So the question may be more along the lines of "is there a fair coin tosser?" not "is there a fair coin to be tossed?" Your instructor may have been referring to a paper of his "Fair Dice." With J. Keller, Amer. Math. Mo., 96:337-339, 1989. (He has it freely available on his website). Just remember that a coin is really just a two sided die.

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Indeed, the problem seems to be more in the tossing than in the coin. The Diaconis example is one where you endeavor to appear fair but really are not. But apparently, people who are not trying to affect the outcome still cannot toss fairly. –  Thierry Zell Nov 1 '11 at 13:51
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Maybe you are thinking of "Dynamical Bias in the Coin Toss" by Diaconis, Holmes, and Montgomery? They show that the same side you started with is slightly biased to come up when you flip an actual physical coin.

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Why the minus?? –  Anon Nov 1 '11 at 15:53
    
While I am the one who down-voted your answer, I note that you only provide a pointer to a paper which is already mentioned with more relevant details in Tom Smith’s answer. –  Emil Jeřábek Nov 1 '11 at 16:04
    
I meant: I am not the one etc. –  Emil Jeřábek Nov 1 '11 at 16:31
    
Ah I see. I didn't click on the link, and since Tom only wrote about the fact that the weighting of the coin doesn't matter, I assumed that is what the paper he linked to was about. Feel free to downvote away! –  Anon Nov 1 '11 at 16:51
    
I re-up-voted you. I also didn't click the link on Tom's paper and I would have written what you did had you not done it. I am pretty sure this paper is the one the OP is referring to. –  Benjamin Steinberg Nov 1 '11 at 17:03
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