A previous question asked about the reality of the gambler's fallacy, in which logic appears to offend common sense. In light of the answers, I am now wondering about the other side of the coin, so to speak.

In physical reality, the "history" of the coin toss would presumably accrue, through friction and wear, into an "unfair" coin. But imagine a physical attempt to make it "perfectly fair." An anti-Laplacian coin. The "coin tosser" must be engineered to absolute consistency... or perhaps absolute inconsistency. Yet if that is the case, the slightest unevenness in the coin would bias it one way or the other, so each toss becomes "ideally" predictable.

So to restore "unpredictability," the coin too must be engineered to an ideal equilibrium of weight and resistance. In this case,however, we are eventually left with an ideal but "faceless" coin and no way of differentiating heads or tails. Indeed the "measured coin" would be no more informative than the "spinning" coin. Again, the results are, for all practical purposes or observers, predictable. Faceless every time.

This is an idealization, of course... as are the tosses in the gambler's fallacy. Yet the result seems to be different. It appears here that the "perfectly random" event or the ideal "fair coin" is a logical contradiction over a history of tosses. Does this cast doubt on the assumptions of randomness in the gambler's fallacy? Which are we to take as the "real" facts in this idealized history of tossing?

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    'Tosser' in colloquial English is like saying 'jerk' in Americanese... Dec 14, 2015 at 15:59
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    @MoziburUllah, more synonymous and as crude as "jerk-off" I'd say.
    – Tom Bowen
    Dec 14, 2015 at 16:06
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    Yes, changed from British self-abuser to abuser of coins only. Possibly a mortal sin as well. Dec 14, 2015 at 17:22
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    The sides of the coin could perhaps be distinguished by putting a tiny (micrometer scale) dot of different colour in the middle of each face. The weight of the dots would have to be equal, and you might want to toss the coin in the dark to make sure there is no Crookes radiometer effect going on. One could certainly get arbitrarily close to a fair but distinguishable coin this way.
    – Bumble
    Dec 15, 2015 at 1:09
  • Yes, the physical problem of a countable difference that "makes no difference" always seems to get closer and closer to an immaterial or "ideal" condition, but I have no idea what the current limit in physics is. Thanks for bringing up this physical paradox... and our modern best answer "arbitrarily close." Dec 15, 2015 at 1:16

12 Answers 12


I'm not sure that I entirely agree with your analysis.

Firstly, and perhaps somewhat pedantically, let's note that although randomness has a precise definition in certain mathematical theories - e.g., information theory, where we define randomness as the inability to compress information - in a philosophical context randomness has no precise definition. Furthermore, it is far from clear that randomness exists in reality. For example, there are deterministic interpretations of quantum theory. So let's stick to unpredictability.

In practice, I do not believe that any unevenness in the coins composition - i.e., the coin having anything other than the ideal centre of gravity - would have any real effect on the predictability of the outcome of an individual toss or a sequence of tosses. This is because other factors, such as the force of the toss and the binary nature of the outcome, would overwhelm any bias attributable to unevenness. The outcome would be completely determined by the original orientation of the coin, the net initial forces exerted, and the characteristics of the landing surface. For any unevenness to play a role in determining the outcome, we would require a near-astronomical number of rotations of the coin, which, of course, is not the case in practice.

Thus, I would argue that achieving an unpredictable outcome for a single toss is possible. If the coin's unevenness does not influence the outcome of an individual toss, it cannot influence the outcome of a sequence of tosses. Any bias that may manifest itself must be bias in those factors that determine the outcome.

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    Thanks. It is helpful to hear about the definitions of randomness. You say unpredictable for a "single toss" is possible, but for a single toss it always is, I believe. You are right to point out all the variables in reality. I was trying to "idealize" all those away to get at the possible paradox in "perfect unpredictability". But I too am not sure I "entirely agree" with my analysis. Dec 14, 2015 at 20:34
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    @NelsonAlexander I was trying to argue - admittedly, not in the clearest of terms - that in practice, any bias an uneven coin might introduce would not effect the outcome of a single toss. Therefore, it could not manifest in any sequence of tosses since any bias effecting the outcome of a sequence must also effect the outcome of an individual toss. The shape of a coin and the binary nature of the outcome are also important considerations. Obviously one could conceive of a large, thin coin which had 99% of its mass concentrated at or near its edge. That is a problem, but impractical case.
    – nwr
    Dec 15, 2015 at 0:55
  • I guess I am driving at the "idealized" or "logical" problem, wherein the "absolute non-bias" or "perfectly equal odds" of heads-tails would necessarily erase any "physical distinction" that would enable observers to establish a difference. The "toss" becomes in a sense tautological. If some physical distinction remained, there is always a chance of prediction. Dec 15, 2015 at 1:11
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    @NelsonAlexander Oh, I see. My reading comprehension is not great. It order to eliminate all bias attributable to unevenness of the surface, oddly enough, it would be only necessary to ensure that the coin's centre of gravity be the "ideal" centre of gravity. This would not exclude the possibility of an uneven surface. It would only require the coin's mass be distributed in such a way the its centre of gravity be located at the ideal point. Such a coin would be equivalent to the flat, faceless coin you describe. I know - I've probably still misunderstood!
    – nwr
    Dec 15, 2015 at 1:21

You can load a die but you can’t bias a coin

Part of your question on the "history" of a coin toss is based on the idea that that 'fairness' - which is generally defined as a lack of bias in an instrument to produce certain results instead of others - is particularly difficult, and you might need to go to great (if not impossible) lengths to produce true fairness.

As the article linked above notes, it is actually remarkably difficult (and for some definitions even impossible) to truly bias a coin. You don't need a uniform surface, a pure elemental metal, or a homogeneous and flat distribution of atoms - a malformed lumpy coin will do.

Slam it down, hit it with a hammer, cover it with any matter or weird marks, impart it with all the history you care to. So long as you don't let the coin bounce as part of your experimental throw, or use a particular 'unfair' coin throwing strategy (like throwing the coin so it never actually flips in the air), you'll end up with a fair coin regardless.

The reason for this is at the heart of what probability means, and in most definitions this is directly a notion of uncertainty. It is not a question of "heads or tails", but rather "it's going to be either heads or tails, and both are equally likely, but we won't know which it is until it has happened".

Experimentally because it turns out that coins are remarkably hard (and with good procedure, all but impossible) to bias, early mathematicians (and sophisticated gamblers) were able to both theoretically and experimentally address notions of uncertainty, probability, risk, likelihood, and the whole field that would later become statistics.

But underneath it all is trying to deal with uncertainty: there is something we don't know. Note that this does not require - or imply - absolute and intractable uncertainty, that we cannot learn more and thus improve our prediction (possibly to 100% accuracy). It just means that given a certain state of incomplete knowledge, we know some things and not others, and some things can be inferred and others cannot.

The notion of 'fairness' is given as an assumption in some mathematical contexts, but in more sophisticated or in-depth treatments of various areas of statistics such a notion is done away with. Indeed, if something is not fair it will - by definition - not behave the same way as an unfair coin, and many techniques of statistics are an attempt to deal with this. A whole mess load of statistical experimental techniques are aimed at attempting to determine whether or not a coin is indeed fair!

Playing with assumptions is at the heart of many statistical methods. If we assume the coin is fair, then we assume certain results are quite typical and others are so unbelievably rare that if we encountered them we would tend to reject the idea of fairness. A coin tossed a 1000 times that always comes up heads, for instance, is theoretically still a possible result of a fair coin! It's just so weird a result that we can comfortably reject the idea of fairness - even though we have not 100% proven anything. This is the basis for "null hypothesis testing", which is quite a useful little tool. One can never insist one knows anything for certain, but we can at least know when the odds are in our favor.

In the end, you don't need a magical randomness box that is impossible to predict. You just need uncertainty. And this is the true cause of the gambler's fallacy: an incorrect belief that you know something useful about predicting the future, when it turns out you just don't. It turns out we humans have a lot of intuitions about how to deal with uncertainty that are off, too. If you are interested in these classes of errors I strongly recommend the books "Against the Gods: The Amazing History of Risk" and "Predictably Irrational", which offer absolutely dozens of other examples about how our intuitive approaches to risk, uncertainty, and probability are weird, mistaken, or just plain wrong.

  • for the most part, i agree with you that "friction and wear" are unlikely to turn a fair coin into an unfair coin. but i don't agree that no physical deformation to the coin will leave it fair. you can bias a coin. i think bending it in a concave manner will make it less likely that the coin lands with the convex side down (since rolling is easier with the convex side down) than with the convex side up. Dec 22, 2015 at 22:31
  • With some practice you can pretty heavily bias the results of a coin flip one way or another, depending on how rigorously define a 'valid' flip. Aug 19, 2020 at 9:00

You can have a fair random toss with photons and two detectors.

First, repeat the experiment M times to satisfy yourself the "coin" is "fair" within some predetermined threshold. Then, pick the experiment result for the Nth photon, were N was predetermined in advance.


The interesting part of this question, at least for me, is that there is no way of distinguishing which side of the coin landed heads.

But nor could you do this when it was between your fingers before you threw it.

You could in principle simply track the motion of the thrown coin with a high-speed camera; and then look at the film at your leisure to see which way up it has fallen.

But this would have to be done continuously for it to have a stable meaning - if the camera 'looked away' for a moment - then one has lost track.

This turns it into a paradox of subject-object epistemology; in the sense of a dichotomy of continuous/discrete observation or measurement which is interesting in a different way.


There's been some studies on this.

US pennies are quite biased. Spinning them, ends up with 80% showing the same face. However, if fairly flipped, the same face shows up only 51% of the time.

Likewise, you can bias the coin by another 1% or so if you control how you flip it.

However, at the end of the day, my answer would be that the question has an assumption in it. It assumes that fair is the same as perfectly fair. If two people are gambling, they can switch their call, and they don't know which side is biased, then it's fair. If the margin of error of fairness isn't scientifically high, then it's fair.

In Australia, by convention, at the start of a sports match the referee flips a coin, and the side is called while it's in the air. Even if one captain knows which side is biased, they are only gaining a 1% advantage, which could hardly be described as unfair. The "fairness" in the coin toss is usually used to state that there's no coin flip trickery (such as feeling the surface, sneaking a peek at the result, or "flipping" it in such a way it only appears to flip.


I must point out that, even if one starts with a biased coin (in the sense that the probability of tossing heads is always P, tails always 1-P, but these are not necessarily equal), it is always possible to synthesize a toss for which heads and tails have equal probability. The procedure is simple: Toss the coin twice in succession; if the two tosses show the same face, discard the results and repeat the process. Eventually one will toss two different faces in succession. Use 'heads' to refer to the sequence heads-tails, 'tails' to refer to the sequence tails-heads. Clearly the probability of 'heads' in any pair of tosses is P(1-P) and the probability of 'tails' is (1-P)P, and these are equal. By repeating until you get an unequal pair of faces, mathematically you will find that the probabilities of each of the two possible face-pairs ends up as 1/2 regardless of the coin's bias.

  • pretty damn close to true, but how could you know that the results of a prior flip didn't alter the probability by some tiny fraction of a percent in between a set of two flips? After all the premise of the question is about imbalance being introduced over time. Aug 19, 2020 at 9:13

Well technically a coin toss is not "random". Toss a coin with the same force in the same direction with the same environment and it will always fall the same way.

The point is that the force with which you spin, the direction and a thousand of little things are "random", and they will randomly interact with the coin.

Now, "random" is the eyes of the beholder at this level; I claim that the force I apply to the coin is more or less random because I don't have perfect control on my muscles. You could say that technically someone could train himself to always use the same amount of force (down to, say, the 20th decimal digit) so that the coin toss is not random anymore, but actually deterministic.

But reality is, at some fundamental level, random; quantum mechanics is needed to even begin to understand this statement(and I'm not claiming I do) but certainly there is some some fundamental randomness in the universe.

To recap: wether something is random really depends on who's judging. With more control and accuracy you can decrease randomness but only up to a certain point, because the universe is at its core random.

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    Your first statement is unsupported, and presupposes determinism. Your later statements are unsupported and self-described as not well founded, and presuppose non-determinism. Dec 14, 2015 at 23:24
  • @thedarkwanderer I beg to differ. Wether we live in a determinisitc universe or not, the philosophical concept does not apply to newtonian physics. The fact that you fall with a certain acceleration is deterministic; and so are most of physics law. Or, the scale at which we observe this phenomena is so large that all uncertainty can be dissipated. At a smaller level quantum effect come into play, but this does not mean that at human (or coin) scale we still have uncertainty about something as simple as tossing a coin and predicting where it lands, given all the necessary informations.
    – Ant
    Dec 15, 2015 at 0:19
  • @Ant. I agree to a certain extent, but not sure this captures the gist of my question. When we attempt to create the "fair coin" even at a photon level, can we "design randomness"? Or even define it? Yet some level of randomness is presupposed in the "laws" of probability. What does that mean? What sort of problem does that indicate? Dec 15, 2015 at 0:27

I believe that the source of the "problem," is the third sentence. "to restore unpredictability, the coin must be engineered to ideal equilibrium and resistance." in which case we are left with a "faceless coin." This conclusion is totally wrong! The only requirement is for the coin to be balanced with respect to a plane parallel to the face of the coin and passing through the c.g. of the coin. This means we can put indentations on each face different but equivalent. For example, I would make a square indentation on one side of the coin, and 4 (1/4 size) indentations on the other side. This way the "balance" is not affected and the faces are distinguishable.


Those that believe in a Creator of everything would tell you that nothing happen by chance.

As per this belief, since everything is designed there is no real randomness anywhere.

All the data suggests that this is true.

You see you or someone else have to make that coin or cast that die. That process, of creation, ensure that no randomness ever came into being.

Slight flaws in the machine that made the coin or in the cast, temperature then as compared to now, heck even slight variance in air pressure due to a plane flying near by all have its effects.

The best we can get towards randomness is having it indistinguishable from the background. Nothing more.


When mathematicians talk about throwing coins, that is just an illustration of principles in probability theory or stochastics.

Real physical events will follow the rules of probability theory well if things that are assumed to be random are unpredictable (for us humans) and behave close enough to being random.

A “fair coin” is just a shortcut for a coin where the outcome is unpredictable to us, and not distinguishable for us from having 50% chance for each side, independent of the circumstances. You don’t need fair coins. You just need to be aware that if you use a coin that is far enough from being fair, your results will tend to disagree with results predicted by theory.


The randomness of a coin toss comes predominantly from the insufficient control of the tossing action.

Any irregularities in the coin composition play no role whatsoever as long as the tossed object meets the definition of a coin.


A fair coin's experimental probability will match the corresponding theoretical probability. So even if the coin's biased (no coin is perfectly balanced), if the two aforementioned probabilities are approximately equal, it's safe to assume the coin's fair. Either that or something really weird's going on.

Randomness is, as was already mentioned, unpredictability and most coins that I've seen tested in mathematical vidoes seem to behave randomly.

I see the contradiction now. To be explicit, no coin is fair (we don't have the knowhow to make one), but coins, when tested, behave like they're fair coins.

The second part of the contradiction requires better/stronger evidence.


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