Bias in coin tosses

When we have to randomly but fairly choose between two outcomes, we instinctively reach for the coin toss. It is because it is an article of faith that the two outcomes of heads and tails are equally likely. But the two sides of the coin are not identical, and hence that slight difference may make a difference in outcome probabilities. In fact, there are four possible forms of bias that may exist. It is possible that either heads or tails may come out on top slightly more frequently or that there is a same-side bias (i.e., the side that is on top when flipped is more likely to be on top when it falls) or an opposite side bias.

Hence this is really an empirical question that can be tested experimentally by tossing coins and counting the outcomes. We know that due to statistical fluctuations, any limited number of tosses are highly unlikely to result in exactly equal number of heads and tails even if the coin is completely unbiased. Statistically, the equal number result for a fair coin is a limit that is approached as the number of tosses gets larger and larger.

The Stanford statistician Persi Diaconis (who is also an expert magician) developed a physics model based on human coin tossing that predicted a same-side bias of 51%. A new study has tested his theory by flipping coins an incredible 350,757 times, a number that is large enough to draw fairly robust conclusions. The result seems to be in agreement with Diaconis’s model, with a slight bias in favor of coin falling with the same side up as it was when flipped. The result was 50.8% with 95% confidence limits of being between 50.6% and 50.9%.

The standard model of coin flipping was extended by Persi Diaconis who proposed that when people flip a ordinary coin, they introduce a small degree of ‘precession’ or wobble—a change in the direction of the axis of rotation throughout the coin’s trajectory. According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started (i.e., ‘same-side bias’).

In order to get such a large number of tosses, 48 of the co-authors of the paper tossed coins of 46 different currencies and denominations to get the huge number of data points. One of them did as many as 20,100 flips while another (clearly a slacker) did just 701. There was considerable variation among the flippers, with the same-side bias ranging from 0.487 to 0.601.

This variability is consistent with Diaconis’ model, in which the same-side bias originates from off-axis rotations (i.e., precession or wobbliness), which can reasonably be assumed to vary between people. Future work may attempt to verify whether ‘wobbly tossers’ show a more pronounced same-side bias than ‘stable tossers’.

So what are the implications for this, especially when it comes to betting?

Could future coin tossers use the same-side bias to their advantage? The magnitude of the observed bias can be illustrated using a betting scenario. If you bet a dollar on the outcome of a coin toss (i.e., paying 1 dollar to enter, and winning either 0 or 2 dollars depending on the outcome) and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average. This is more than the casino advantage for 6 deck blackjack against an optimal-strategy player, where the casino would make 5 dollars on a comparable bet, but less than the casino advantage for single-zero roulette, where the casino would make 27 dollars on average. These considerations lead us to suggest that when coin flips are used for high-stakes decision-making, the starting position of the coin is best concealed.

So there you are. Science news you can use for profit!


  1. birgerjohansson says

    If you read the anthology by Terry Pratchett that was the first Discworld book, you know that areas that were devastated by Mage wars have a third option; the coin landing on edge.
    These places are just as dangerous as the temple of Bel-Shamaroth were even Conan wwould hesitate to go.

  2. EigenSprocketUK says

    I wonder if there is another potential form of bias: one which develops for an individual otherwise-equal coin during many tosses. I could imagine that slight bends and edge dents might develop biased on a slight random outcome from the first few, for example.

  3. Pierce R. Butler says

    Back in the ’60s, IIRC, somebody did some experiments with a machine tossing a coin, reading the result, and then either delivering a shock, or not, to some cockroaches in a cage. Their stated goal was to see whether the potential psychic powers of the bugs would reduce the number of shock-inducing results -- but it turned out the flips produced (slightly) significantly above 50% shocks.

    The experimenters pronounced themselves baffled by this outcome. Reading about it, I hypothesized at least one member of the team may have had a psychokinetic power greater than that of the insects, and really hated cockroaches.

    Perhaps the IIRBs halted further experiments of this kind, or maybe I just missed the follow-up tests, if any.

  4. says

    Assuming that this result can be replicated and is accurate, we reach the given concluding example:

    If you bet a dollar on the outcome of a coin toss … and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average.

    This would take several hours to do, meaning that you’d be making less than minimum wage. Also, you have to have considerably more than a dollar to bet because at points you can be several times the betting amount “in the hole”, and that means that you cannot solve the minimum wage issue by just issuing a much larger bet (i.e., what else could you do with that capital). Besides, as long as the person calling it doesn’t know the starting point, it’s close enough to 50/50 that it won’t matter. Now, if there was a specific heads or tails bias, that would be another issue.

    The thing that I find most interesting is why we even call it “tails”. One side almost always has a person’s likeness on it, so “heads” makes sense, but the other side seldom has a likeness of a tail. And while a tail can be seen as the opposite of a head for many animals, that certainly is not the case for humans. If you ask most people what the opposite of the human head is, they’d probably say “feet”. I don’t ever recall seeing a coin with a head on both sides* (or feet on one side for that matter), so maybe we should just call it “heads or not”.

    *The Buffalo nickel has a bison on the reverse, so it does have two “heads”, but one of those heads also has a tail. I guess the lesson then is to replace the flipping coin with a buffalo nickel and call “heads”, so that even if it comes up tails, you can argue that there is still a head on there.

  5. VolcanoMan says


    The opposite of a human head may be, to most people, their feet, but in medicine, the word used to indicate that something is towards the head of a person is “cephalad” (which makes sense), but to indicate the other direction, we use “caudad” (which actually does mean “towards the tail”). So heads vs. tails does indeed make sense from this perspective.

  6. Jazzlet says

    jimf @#5
    Several British coins have the royal lion(s) and/or Britannia on the reverse -- all coins have the monarch’s head on the obverse side.

  7. jrkrideau says

    After reading some responses here I have to ask what coin was tossed?

    If the results hold true will we get all those @#$% statistics texts to stop using )((**& coin tosses?

  8. sonofrojblake says

    I once asked a colleague in the tearoom, during a discussion of statistics and probabilities in test results on the chemical plant we worked on, what he thought the odds were for a coin toss. He was a recent chemistry graduate, and he opined it was 50/50. I took a coin from my wallet, called tails, and tossed it. It came up tails. I asked him what he thought the odds were now of heads coming up. Still 50/50 was his reply. I called tails again. It came up tails. I tossed it TEN TIMES, each time correctly calling tails, each time asking him if he thought the odds were now in favour of heads coming up, and he stubbornly persisted in saying the odds of the next result were NOT in any way dependent on the result of the previous one -- that heads had a 50/50 chance of coming up, even after a run of tails. After ten successful calls of tails -- odds in the region of 1000 to one against -- I did reveal to him that the odds of heads coming up were in fact not 50/50, but zero, as the coin I was using had tails on both sides. Lesson -- if the results aren’t what you expect, check there’s not something screwy going on.

  9. Silentbob says

    When I was a youngster, the classic way of tossing a coin was you caught it in the tossing hand and slapped it face down on the back of the other hand. When you lifted the tossing hand away, that was the result. So the bias would have been reversed.

  10. sonofrojblake says

    @jrkrideau, 9:

    After reading some responses here but not reading the original post properly I have to ask what coin was tossed?

    Fixed it for you.

    From the OP:

    In order to get such a large number of tosses, 48 of the co-authors of the paper tossed coins of 46 different currencies and denominations to get the huge number of data points

    There’s surely work to do to find the fairest and screwiest coin in the world.

    The first coin I ever remember being tossed was a British 1970s ten pence piece, which had a lion on the obverse with a very obvious tail, so I never questioned it.

  11. Silentbob says

    @ 10 sonofrojblake

    I hadn’t read this comment before posting, knowing sonofroj to almost never be worth reading, but having read it, I must say…

    Your story is that you had a collegue who trusted you, and correctly called odds of a coin toss, but you were a liar and cheater who had a double sided coin (something only an inveterate conman would have), and despite you lying and cheating several times, you innocent collegue continued to trust you…

    … and you think this reflects badly on your coworker?!!!

    Holy fuck, man, what is wrong with you? X-D

    Did it occur to you most people don’t assume others are inveterate liars, cheaters, and conmen?

  12. sonofrojblake says

    Ah, Violentblob, so good to know you couldn’t resist reading what you characterise as almost never worth it -- you may wish to address your obsessive behaviour with a therapist or something.

    you were a liar and cheater

    I was neither. I absolutely did not lie about the coin -- I’m VERY careful with my language when doing this sort of thing. And I cheated him out of nothing at all -- nothing was riding on the coin tosses. He made an incorrect assumption about the coin, a perhaps reasonable one… except unlike you, he knew me. Specifically, he knew me to be an occasional perpetrator of cheap tricks (here, watch while I make this little hanky vanish…). So the assumption the coin WASN’T screwy was not as reasonable as it might have been if he’d been standing in front of a more boringly predictable person. You, for instance.

    Double headed coins are not the exclusive preserve of the inveterate conman, although I’m unsurprised your “imagination” -- and those are VERY heavy quotes -- can’t come up with another reason to carry one, even as one has been provided right in front of your face. (Clue: entertainment.)

    you think this reflects badly on your coworker?!!!

    What on earth makes you believe I think that? I certainly didn’t say anything to that effect, not least because I don’t think it. Certainly nobody in the room thought any the worse of him for it, I made sure of that. I did use the word “stubbornly”, yes, but on one level I absolutely applaud his dedication to the observation that the gamblers’ fallacy (that the odds will “correct themselves” by making heads more likely after a run of tails, say) IS FALSE. And I absolutely own up to “cheating” to get an effect, but magicians do that all the time. But the point -- that if something doesn’t behave as you expect, if it doesn’t seem legit, start looking for something you’ve missed -- was well received.

    Not by you, obviously, but then it’s been established time and again here that you’re a fucking idiot.

  13. KG says

    Tom Stoppard’s play Rosencrantz And Guildenstern Are Dead (kind-of Hamlet from the p.o.v. of these bit-part losers in Shakespeare’s play) begins with the two repeatedly betting on coin tosses -- they apparently think the coin is a fair one. It comes up heads 92 times in a row. Guildenstern (who’s been betting on tails) then suggests that they may be “within un-, sub- or supernatural forces” -- which of course, they are: Stoppard’s script.

    On another tack, I’ve just been reading Marcus du Sautoy’s The Music of the Primes, which is about the Riemann Hypothesis, a conjecture in number theory concerning the distribution of prime nmbers (actually, it’s more a series of potted biographies of the mathematicians who’ve tried to prove it than an exploration of the hypothesis itself). One thing that grates on me somewhat is the du Sautoy repeatedly raises the idea that the distribution of primes may be “random”, explicitly comparing it to a sequence of coin tosses -- but that’s one thing it absolutely cannot be! Whether a number is prime or not is a necessary fact, true in all possible worlds, far more fundamental to the nature of reality than any mere physical constant.

  14. sonofrojblake says

    Hmm… depends on your definition of “random” I suppose.

    On one level, the distribution of primes is, as you rightly point out, the hardest and most deterministic of facts, arguably true and identical across any possible universe. It’s true BY DEFINITION -- numbers be numbers.

    But… given the gap between this prime and the last, can you *predict* how large the next gap will be? As in -- can you tell me what the next prime number will be without first going through the business of testing each number in sequence to see if it’s prime or not? Because if you can’t, that’s at least somewhat analogous to be unable to predict the outcome of a coin toss, and instead having to actually toss the coin and see what comes up.

  15. KG says

    Hmm… depends on your definition of “random” I suppose.

    Well if by “random” you don’t actually mean random, I guess they could be “random”. In a random sequence, each member must be produced by a process independent of the preceding members. The pseudo-random sequences often used in computer applications are called pseudo-random because that’s not the case. It’s cetainly not the case for primes, as Eratosthenes pointed out a couple of millennia ago.

  16. sonofrojblake says

    But… I’m struggling with this, to be honest.

    Pseudo-random sequences can be predicted if you have the algorithm and the seed. If you’re saying that prime distribution is NOT random, but in fact pseudo-random, then aren’t you saying that there definitely IS an algorithm and seed which, if you knew it, in principle would allow you to perfectly predict the distribution of prime numbers. Since it still seems to be something of a big deal when a new “biggest prime number” is discovered, then there are presumably two possibilities:
    1. there’s an algorithm, but we don’t know it -- because if we did calculating the “next” prime would be trivial OR
    2. there’s no algorithm.

    In either case, though, since we don’t have the algorithm, whether it exists or not, isn’t prime distribution FOR PRACTICAL PURPOSES random? (I know that’s not the same as actually being properly random, but I’m an engineer, not a mathematician).

  17. friedfish2718 says

    This coin flip issue reminds me of the question: Given a circle of radius 1, what is the average length of a random arc? The question is incompletely defined for it did not mention how one goes about drawing (randomly) the arcs.
    How random is the flipping process itself? Answer: not quite random.
    I can build a robotic coin flipper, with each flipping process being repeated in exquisitely exact manner time after time: the robot can be programmed such that if the pre-flip coin is Heads, 99.99% of the time the post-filp coin is Heads. Or 99.99% of the time the post-flip coin is Tails.
    A human is not a robot (that is for sure). There is some variability in the human dynamics (if one focuses on 1 human) and quite likely the variability is narrow enough to deviate from the 50/50 ideal results.
    When I was a kid, I would hold a coin (on a table) on the edge, with Heads facing me. I flick the coin. The coin is spinning on the table on its edge. The coin spin slows down and eventually it lands flat on the table. About 70% of the time, Heads facing up. It is a skill. I was more successful with large coins than with smaller ones.

Leave a Reply

Your email address will not be published. Required fields are marked *