# Is tossing a coin a fair way of randomising a group into two groups?

So myself and my uncle are having an argument over whether a coin flip is truly random. I argue it isn't because in real terms a coin tosser will always manipulate a coin so the result isnt 50/50 therefore it is not a good choice as a randomisation technique for assigning groups in clinical trials. However he argues that it is the minute imperfections in the coin tossing that creates the randomness. So then he postulated a machine that would eternally be able to toss a fair coin and have it land on heads and to be honest I just need someone to settle this argument for me. Is tossing a coin a fair way of randomising a group into two groups?

• I think this depends on the coin. If it is a fair coin, the tossing is appropriate for assigning groups. May 19 '15 at 19:33
• Thanks for your reply. The point I was trying to make was that, as no coin tossing person can say they tossed the coin in such a way that there is a definite 50/50 chance of heads/tails then it is not really random. Not sure if I am right though May 19 '15 at 19:36
• @random_guy if one thing could be known for sure is that it does not depend on coin (if it has both sides different), Gelman et al. wrote on it, see my answer.
– Tim
May 20 '15 at 8:51
• Also note that usually you dont want random assignment on patient-by-patient case but a random split. You dont want to do a 8-2 study instead of 5-5 because your coin just fell that way.
– Erik
May 20 '15 at 10:51
• Your uncle probably has seen Persi Diaconis' coin-tossing machine! May 21 '15 at 7:34

Yes, coin flip is truthfully random process. While it is possible to load a die, so that it favours certain outcomes, you cannot bias a coin (see paper by Andrew Gelman and Deborah Nolan published in The American Statistician for further details). You can argue that coin toss is a deterministic process and in fact you can build a mathematical model that describes the process, however its outcome is random. For learning more on physics on coin toss check lectures by Santosh S. Venkatesh on Probability course on Coursera.org where he describes the dynamics of coin toss in detail and provides an argument why it is truthfully random (Tableau 7), you can also check Keller's paper The Probability of Heads and short paper by Mahadevan and Hou Yong titled Probability, physics, and the coin toss). The deterministic process like this could be random because it is a kind of process where small changes in the initial parameters (velocity, angular velocity etc.) make a huge difference in the outcome, what makes its behaviour chaotic (check lecture by P. Diaconis titled The Search for Randomness).

Actual experiments have shown that the coin flip is fair up to two decimal places and some studies have shown that it could be slightly biased (see Dynamical Bias in the Coin Toss by Diaconis, Holmes, & Montgomery, Chance News paper or 40,000 coin tosses yield ambiguous evidence for dynamical bias by D. Adolus). Diaconis et al. reproduce a histogram of one of such experiments where 103 students tossed coins each 100 times (see below).

Notice that in real life people throw coins with different strength, at different height, start with holding coins lying on their hands with different angles, catch them at different time and in different way, atmospheric conditions differ etc., this makes the actual outcomes vary between coin tosses and coin tossers as in the picture above.

As A. Donda and Glen_b noticed, there were examples of people who learned how to purposefully throw coins to get certain outcomes and Diaconis et al. managed to build a coin tossing machine that could toss coin for a certain outcome.

Does this all make coin toss not reliable? Washington Post quotes one of the authors of Diaconis et al. paper:

I asked Holmes whether coin flips used for, say, football, should be eliminated because they are biased. The answer is no, as long as the person calling the flip doesn't know how the coin is going to start out. In football, the tosser is never the caller; the tosser is supposed to be a referee. But if you are both the caller and the tosser, well, that changes things. Knowing about the bias in coin tosses give you an edge, albeit a tiny one.

Also biases observed in the experiments in most cases are not really greater then what we would expect from random draws from Binomial distribution (see plot below), they vary between experiments and coins used. In most cases they fall into the 95% highest density region of binomial distribution parametrized by $p=0.5$ and sample size equal to total number of coin tosses in the particular experiment (i.e. we would expect 95% cases to be not more extreme then this). In two cases results fall outside the interval: in case of tosses by Janet (described by D. Aldous) and tosses by Robin's students (as described in CHANCE News). However it is hard to compare the experiments due to differences in used methodology (single tosser vs multiple tossers, single coin vs multiple coins etc.) and methodological flaws (e.g. in Robin's class case students were tossing the coins outside of class, so it was not monitored how carefully did they follow the instructions).

On the plot above we see proportions of heads in different experiments with accompanying 95% highest density region. The results are gathered from Flipping, spinning and tilting coins paper in CHANCE News, Coin Tossing and Spinning - Useful Classroom Experiments for Teaching Statistics paper by Helmut Kuchenhoff, and results from experiments by D. Aldous. Ball sizes reflect the sample sizes used in the experiments. On the $x$-axis we see the outcome (proportion of heads), and on the $y$-axis cumulative probability of outcome smaller or equal then the one observed as calculated from the binomial distribution.

Notice however that in most real life cases you do not need truthfully random values, but rather you are interested in numbers that behave like random numbers. No matter if you are doing statistics, or you implement a cryptographic algorithm to encrypt data, what is used for those purposes is pseudorandom number generators, i.e. deterministic algorithms that produce output that is hardly distinguishable from truthfully random values. This is enough even for cutting-edge cryptographic algorithms.

So summarizing, the research in this area gave mixed results and what can be said for sure is that there are multiple factors that influence coin toss. The answer for your question is yes, coin toss is random because it provides enough randomness to consider its outcome random.

Quote from E. Borel by Bruno de Finetti in his paper Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science can serve as a motto for this answer:

"One can bet, in heads or tails, after the coin, already tossed, is in the air, so that its movement is determined. One can also bet after the coin has landed, on the sole condition that one does not see on what side it has landed. Probability does not lie in the fact that the event is undetermined (in the more or less philosophical sense of the term) but only in our inability to predict what possibility will take place, or to know what possibility has taken place."

• The Numberphile argues the opposite. May 19 '15 at 21:01
• Since Persi Diaconis (among others) can demonstrably get heads seemingly at will, the assertion that you cannot bias a coin is in effect completely missing the point. A human can take an ordinary coin (which we might regard as symmetric - and so presumably 'fair' - for practical purposes) and achieve a particular desired outcome on the coin, almost always. Perhaps it is possible you can't bias the coin but you can certainly bias the process of the toss itself. May 20 '15 at 2:58
• Just in case anyone wants to see it, a young Diaconis demonstrates it here May 20 '15 at 8:57
• Good update, +1. Btw., an imho really good discussion about what's "really random" including a section on "How to cheat at coin and die tossing" is given by Jaynes and Bretthorst in Ch. 20 of Probability Theory. The Logic of Science called "Physics of 'random experiments'". May 20 '15 at 16:23
• @amoeba done, thx
– Tim
May 2 '17 at 12:08

Regardless of how fair the coin toss is, it is not a good way to assign treatments in a clinical trial. With a perfectly fair coin toss, it is possible that all of the subjects get assigned to the same treatment! While that would be rare, it would be fairly common to end up with a very lopsided distribution of treatments.

Better: Shuffle the order of the subjects, and give one treatment to the first half and the other treatment to the other half. Or write down treatments on n cards (half one treatment, half the other), put in a hat, shuffle, and draw one card for each subject (of course this can be computerized).

Bottom line: You want to randomly assign subjects to treatments, not randomly choose a treatment for each subject.

• Apologies for off-topic ping: there is a suggestion on Meta to make [randomized-experiment] a synonym of [random-allocation] tag (stats.meta.stackexchange.com/a/4651). You have enough reputation in this tag in order to vote for this suggestion here: stats.stackexchange.com/tags/random-allocation/synonyms - it now needs 4 upvotes to go through. If you disagree with the proposal, consider commenting on Meta to explain why. I will delete this comment soon. Cheers. May 2 '17 at 12:05