There Is No Such Thing As A Fair Coin

There Is No Such Thing As A Fair Coin

Not only is there no such thing as a “fair” coin, there are no “fair” dice, either. Or “fair” anything. Which I shall prove to you.

From Khan Academy, a popular, and wrong, view:

What does it mean to say you are flipping a fair coin?

Fair means that the coin has a 50-50% chance of getting HEADS or TAILS. They have that because in some other situations, there are “unfair” coins that have more chance of getting one result or another.

If the evidence we have is “Here is a coin with two sides, H and T, that when flipped must come up one side or the other” the probability of H given this is 1/2. We did not have to add “fair” to this. What would adding “fair” do? Make it even more 1/2? The idea does not make sense.

For comedians who ask about coins landing on their sides, please read the evidence again more carefully.

Our coin example is much the same as this: “Here is a device that must be in one of two states, H or T”. The probability of the device being in state H given this evidence is also 1/2. Here there is no way to meaningful add “fair”.

How about evidence like this? “Here’s a unfair quarter, with two sides, etc.” What are the chances, given this evidence, it comes up H? The best we can say is “not 50%.” But that’s adding to the evidence the implied meaning of unfair and the tacit knowledge that something will go wrong, we know not what. Indeed, there are so many possibilities for tacit and implicit premises here it’s difficult to be coherent.

Coins do not have probabilities. Flipped coins do not have probabilities. Nothing has a probability. Probability is only defined by the evidence you assume. Probability is all in your mind.

Ask my old advisor Persi Diaconis to flip a quarter. What is the chance it comes up H? Well, to you, it is 1/2, if you used something like that evidence above.

But to Persi, who has a coin flipping machine, the probability is 1. The coin will always come up H. I have a fuller description in the talk I gave in Phoenix earlier this year.

The machine was built to control the causes the coin comes up one way or the other. Those causes relate to the physical aspects of the coin, the spin, and the force it’s given. If you know the relevant causes, all of them in all aspects, then you know the outcome.

It’s a simple as that. In every problem. Really.

We don’t need “long runs” or any other such nonsense, which speak of infinity, about which nobody knows anything in any empirical way. Probability is not frequentism. In frequentism no probability can ever be known until infinite trials are conducted. Lots of luck with that. Especially in counterfactual trials which cannot be conducted, but in which we can easily see probabilities. Instead, probability is the degree of truth propositions have given assumed evidence. Probability is logic.

Now for the proof there are no “fair” coins.

When you think of “fair” coins you likely have an idea of the symmetry of the coin. It’s round and of roughly even thickness. But that’s not real symmetry. It’s only approximate. An American quarter has unequal mass across the obverse and reverse. The grooves on the edge are not perfectly equal. So this is not a “fair” coin if symmetry is the definition.

But maybe you think you can manufacture such a coin, taking great care the mass is as uniform as you can get it, along all the coin’s aspects. Alas, you will not succeed. And you won’t be able to know, either. Are you perfectly certain the number of quarks, or whatever, are equal in number across every dissect? No, sir, you are not. And cannot be.

Symmetry fails, then, to describe “fair” coins. (Though, of course, if such a thing existed, knowing this would not change the probability we first computed.) But even supposing an absolutely symmetric coin existed, it still has to be flipped. Then what?

The nail that flips it has friction, which varies. The air in which the coin flies has varying densities. The force exerted from other objects during the flip, like the earth, are varying. The spin varies. The power given the coin, and the location it is given on the coin, varies.

Just what is symmetry supposed to mean in all this?

Well, nothing that is coherent. But we do know that if we control all the causes, at least to an important degree, we can get the coin to do what we want.

So what about your coin in the special weird circumstances you create? The probability we began with is a model. Start with that (your “prior”), and then take evidence and see if your model is useful or not. Or make a new model using that evidence.

There are no such things as “fair” coins, and no such thing as probability.

If you’ve read this far, congratulations, because this has much more importance than you might have thought. Believing probabilities exist makes bad science. Or at least limited and incomplete science. Quantum mechanics, anybody?

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  1. Hagfish Bagpipe

    A local quantum mechanic looked over my Honda recently. He said it had a fair probability

    ”Quantum mechanics, anybody?”

    Yeah, what about those shady mechanics? In their greasy coveralls, toting around those stupid little packets, bunch a jokers, never knowing where or what they are, here, there… anywhere? A fair probability just another bogus branch of The Science, amirite, Doc? Might be fun see you give them quantum mechanic boys a good caning.

    Had one a them quantum mechanics look over my Honda recently. “Fair probability she’s gonna flip you in the ditch, because causes”,

  2. Hagfish Bagpipe

    That last bit was a discarded wisecrack. Only the top drawer stupid is supposed to make the cut but the quantum boys make stuff appear out of nowhere.

  3. Nice.

    One quibble: I’m in the “a pox on both houses” camp in re frequentist/baysian. As i see it P(H) is either 1 or 0; the estimate P(H)=0.5 says I have no information favoring either 0 of 1 over the other. So 11011010X1000 is perfect encryption if P(1)=0.5 for any X in the sequence.

  4. Incitadus

    The coin toss is the introductory lecture to Stats 101, the coin toss is the fiction that
    keeps Las Vegas alive. Reality is shaped and directed by a working knowledge of Magick
    and the certainty that humans can be made to believe anything with enough time and
    repetition. ‘Magick’ you scoff as you watch in horror the androgynous prototype reemerge
    from chrysalis. All that is old is new again Thelema…

  5. If there are no “fair” dice, then there cannot be “loaded” dice either … and if you believe that, I invite you to come join in a high-stakes dice game using my dice.

    If I throw a single die 10,000 times, and half the time it comes up with the number “1” on the top, is that a “fair” die or a loaded die?

    “Fair” simply means that the outcome of actual testing is within some calculable amount of the probable outcome based on statistics. For a single die, that means it should come up with any given number 1/6 of the time, within a statistical deviation of course. And if it comes up with “1” half the time, it’s definitely not a “fair” die.

    So. Is my computer random number generator “fair”? Here’s a test:

    > thethrows=sample(1:6,100000, replace=T)
    > length(which(thethrows==1))/100000
    [1] 0.16722
    > binom.test(length(which(thethrows==1)),100000,p = 1/6)

    Exact binomial test

    data: length(which(thethrows == 1)) and 1e+05
    number of successes = 16722, number of trials = 1e+05, p-value = 0.6377
    alternative hypothesis: true probability of success is not equal to 0.1666667
    95 percent confidence interval:
    0.1649118 0.1695475

    Since the result (0.16722) is well within the confidence interval, we can say that it is indeed “fair”. And if it came up with “1” half the time, it would assuredly not be “fair”.

    I fear you’ve become lost in statistical minutia and are ignoring the plain meaning of words.


  6. Cary D Cotterman

    I dropped a nickel once and it came to a stop balanced on its edge. I predict there is extremely low probability that this will ever happen to me again, no matter how many nickels I drop, but I suppose the probability of it happening again is still exactly the same as it was before it happened the first (and probably only) time.

  7. Michael 2

    “Fair” seems to mean I get something for nothing. My teenage stepson argued it would be “fair” if 6 year old washed the same number of dishes as he. It would have a certain type of equality but does not seem “fair”. Fair is measured by different people in different ways. Flipping a coin or tossing dice is *deterministic* even if you don’t know all the determinants.

  8. The True Nolan

    “Nothing has a probability. Probability is only defined by the evidence you assume. Probability is all in your mind.”

    Exactly, and thank you for repeating this concept enough times that it FINALLY soaked into my impervious skull. A probability is not a THING. It is not even a quality of a thing. It is (at best) an abstract statistic and thus “exists in your mind”.

    I would say the same thing about averages. The Earth does not have an average temperature. It may have temperatures here, there, and everywhere, but we certainly do not know very much about what they may be or how they are distributed around the world. But an AVERAGE temperature? Ain’t no such thing. It is (at best) an abstract statistic and thus “exists in your mind”. In the case of Global Warming Cultists, the mind in which it exists is mostly empty… and not in a desirable Buddhist fashion.

  9. I’m with Eschenbach, “fair” is the condition where the rubes won’t complain about being bent over and shafted hard from behind while their money is being stolen. As for probability, try explaining to the people in line for lottery tickets then you’ll see it obviously makes no sense. Except if you’re a lawyer, in which “fair” means something not remotely related to any of the normally assumed meanings of the word and is defined for the purposes of bending you over and shafting you hard from behind while your money is stolen.

  10. Well, if you want to be OCD anal, the ACTUAL answer to your questions given evidence is not “50%”, it’s “I don’t know”. Nowhere in the evidence does it say the distribution of possible outcomes is flat. To say “50%” is to merely posit ONE of (infinitely) many possible ways the probability can be distributed across possible outcomes. People normally assume the question is related to the real world and in real world properly tossed coins do in fact tend to distribute their outcomes equally across options. But there’s no necessary fundamental principle that would require them to. Yes, you can accuse me of the Deadly Sin of Reification, but you still won’t be able to prove an abstract two-state machine that can change it’s state randomly will HAVE to end up being equally distributed between those two states. 🙂

    I also wrote an overly long post on coin flipping and the limits of science given randomness:

  11. that someone cannot write a two-state machine that proves the case is empirical evidence; got it.

  12. Pk

    Similarly, replace the coin with a vexxine filled syringe. There is no safe vexxine. At least not unconditionally.

  13. James J. Roper

    It’s a whole nonsense argument about a fair coin – it’s just a way to describe a binomial process, not a comment about coins per se. Binomial distributions are useful for examining many processes with two possible outcomes. The whole rant was meaningless, as far as I can tell.

  14. Gunther Heinz

    The probability of a coin landing on its edge is 1/6000. So total probabilty is 1/2 + 1/2 + 1/6000. This is known as an improper fraction because you can’t just add.

  15. Nate Winchester

    I can’t see anything analyzing coin tosses without thinking about the infamous No Game No Life coin flip scene…

  16. Nate Winchester

    But maybe you think you can manufacture such a coin, taking great care the mass is as uniform as you can get it, along all the coin’s aspects. Alas, you will not succeed. And you won’t be able to know, either. Are you perfectly certain the number of quarks, or whatever, are equal in number across every dissect? No, sir, you are not. And cannot be.

    This reminds me of the story of how a small change to the thread STICHING baseballs together was shown to have an impact on home runs in a game.

  17. Milton Hathaway

    Hagfish – it’s a bit depressing that your bottom-drawer witticisms exceed the amenities of my best mental furniture. I guess I should stop shopping the thrift stores.

  18. Rudolph Harrier

    The appeal of the frequentist definitions is that they make the math solid, since they reduce everything to tautologies. If I flip a fair coin “infinitely many times” will the average number of heads converge to 50%? Why, yes, because that’s precisely what we’ve defined “fair coin” to mean.

    But the disadvantage of this approach is that we cannot identify a fair coin in reality. We cannot see the limit, only finite observations. Even if we get a billion heads in a row we still cannot say that the coin is not fair, as there might be a billion tails to come that correct for that in the limit. Now you may say that in a truly fair coin we would have a more even distribution of heads and tails. But our frequentist definition of “fair coin” says nothing about the distribution, only the limit, so we are introducing new conditions midstream. And if we DO say that a fair coin must not have unlikely streaks, then we are actually talking about a different, non-independent, random variable. For if we do not allow the coin to have a billion heads in a row, then if it shows 999,999,999 heads in a row (possible for a truly independent random selection) then the billionth flip must come up tails, and hence the billionth flip is not independent of the first 999,999,999 flips.

    But suppose you insist we be less pedantic. Maybe you say “c’mon, if we flip the coin a thousand times and it’s within 5% of a 50/50 distribution we’ll call it fair. Don’t worry about the philosophical justifications.” Okay, but who is flipping it? As Briggs notes there are machines which can flip coins to a desired result with near perfect accuracy, and there are people who can obtain similar results less reliably. If you say “well, we’ll have someone who doesn’t cheat flip it,” then how do you test for that? That is, suppose someone does flip the coin repeatedly and gets 80% heads and 20% tails, do you conclude that the coin is weighted, or that the person flipping it was cheating, some combination of these factors, etc.?

    There could be factors we haven’t even dreamed of. Perhaps the coin is made of different metals of different compositions in such a way that its weight is balanced but the coin is more likely to land on one side rather that the other in the presence of a magnetic field. Then we not only have to worry about cheating in our tests, but also magnets. And there could potentially be many strange confounding factors like this, putting us into a frame problem.

    So it is difficult enough to determine if a coin flip is “fair” because all confounding causes must be controlled. But of course we assume fair distributions in much more complicated scenarios, where it is all but certain that there are causes we are unaware of and can’t control for. How nice it is to be able to fall back on the tautological definitions in our statistical philosophy to distract us from the impossibility of actually finding those objects in reality.

  19. No such thing as “radiative forcing” either.

    Radiative Forcing is defined as the difference between two values of outgoing longwave radiation, OLR. One OLR value is measured at the top of the atmosphere, ToA, as it leaves for space. The other value is assumed (by using the Stefan-Bolzmann Law, for example, a calculation of OLR emitted immediately above the (land and ocean) surface.

    The problem here is that the Stefan-Boltzmann Law is illegitimately used. It only applies to black bodies. Black body cooling only makes sense in a vacuum. Look back at Stefan and Boltzmann’s experiments to determine their law. Every experiment of theirs’ is done in a vacuum. Because if a solid black body, for example, is in proximity with a gas – that gas will do nearly all the surface cooling. At earth surface pressure the cooling due to conduction and convection is over 200 times that of radiation. So the self-styled climate consensus model nearly all earth surface cooling as radiative when, in reality, it’s nearly all conduction/convection!

    See Tom Shula:

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