# Is a coin toss fair?

I've just seen a referee use a coin toss to decide who will start a match which got me thinking, coins aren't perfectly symmetric. Our euro coins have the value on one side, a country specific image on the other side so does this change the heads/tails chance?

Leaving the possibility that it lands on its side out, is there a side I should bet on to have more than 50% chance to win?

• There are too many unknowns: What material is the coin made of? Does the coin land on a grass field, a hard court floor, or the referee's hand? Is the referee tossing the coin fairly? Does one of the players get to choose the outcome once the coin is in the air or is it assigned prior to the throw? Aug 31, 2011 at 0:06
• If you have a specific application, you could try collecting statistics. I don't think there's another way to do it. Aug 31, 2011 at 0:50
• The idea of "fairness" is a human construct. I know it's nit-picky, but I think what you mean is "is a coin toss truly random/unpredictable?" Aug 31, 2011 at 16:50
• `Fair coin` and `fair dice` are used terms in statistics. A coin or dice can be truly random but unfair. Or the game is unfair: You win, if you reach a series of 10x head, else I win. It is random, but unfair. Aug 31, 2011 at 21:38
• @user unknown Glad you said that, now the payoff of the joke... alright, fair enough :) Sep 1, 2011 at 5:26

Is there a side I should bet on to have more than 50% chance to win?

No, the asymmetries do not affect the fairness of the coin, if it is caught in the hand.

I draw this conclusion from: You Can Load a Die, But You Can't Bias a Coin, Andrew Gelman, Deborah Nolan. The American Statistician. November 1, 2002, 56(4): 308-311. doi:10.1198/000313002605. Full Paper

Dice can be loaded—that is, one can easily alter a die so that the probabilities of landing on the six sides are dramatically unequal. However, it is not possible to bias a coin flip—that is, one cannot, for example, weight a coin so that it is substantially more likely to land “heads” than “tails” when flipped and caught in the hand in the usual manner. Coin tosses can be biased only if the coin is allowed to bounce or be spun rather than simply flipped in the air. [...] We explain this phenomenon by summarizing a physical argument made in earlier literature.

As well as repeated experiments with students, they use a simple physical model to show that - as angular momentum is conserved - any coin will spend half of its time heads-up and half tails-up. If the coin is spun or allowed to bounce, this model falls apart.

Note: Fake double-headed coins exist. [Ref] Also, magicians exist: Sleight of hand can be used to make the original coin vanish or for the result to be read before the coin is revealed. [Sorry, only anecdotal evidence: I can do that. I can't quite manage to force a result based on the call in the air - I can't think fast enough to make it look fluid - but I see it as feasible with practice.]

• Are there any proofs provided by that reference, other that saying 'No, you can't bias a coin'? I find it very difficult to believe that altering the shape and weight of an object has zero affect whatsoever.
– Rob
Aug 31, 2011 at 2:52
• @Rob, I added a reference to the full paper and a brief summary of the physics they use. They use the very fact that the result is surprising to interest students in the importance of careful testing. Aug 31, 2011 at 3:59
• The simple conclusion is that the paper must be wrong, because it is possible to bias a coin toss. The problem is that you are able to toss it in such a way that it looks like it's rotating around a horizontal axis, but actually the axis is almost vertical and the coin is only slightly declined. This means that the coin very likely falls on the side that was top at the beginning, especially if you minimise the chance it flips after the impact, e.g. by catching it in hand or by letting it drop on a surface that absorbs the kinetic energy easily.
– yo'
Feb 7, 2013 at 10:30
• Thanks, @tohecz: That technique fits under the "sleight of hand" category. It is also covered by Lev Bishop's answer. Feb 7, 2013 at 14:19

I am able to toss any ordinary coin so that it gives my chosen result with about a 90% success probability, and so that it looks like an ordinary toss to an observer. The technique is easy, based the physics of rotating objects, and you can learn it yourself in the late Ed Jaynes excellent book "Probability Theory: The Logic of Science" in chapter 10 "Physics of 'random experiments'"

Therefore in order to know which face will be uppermost in your hand you have only to carry out the following procedure Denote by k a unit vector passing through the coin along its axis with its point on the heads side Now toss the coin with a twist so that k and n make an acute angle then catch it with your palm held at in a plane normal to n On successive tosses you can let the direction of n the magnitude of the angular momentum and the angle between n and k vary widely the tumbling motion will then appear entirely different to the eye on different tosses and it would require almost superhuman powers of observation to discover your strategy

So, to answer your question, you should bet on whichever side you think the referee desires to favour to win the toss.

• I think that the question was referring to the coin shape, but interesting nevertheless... Aug 31, 2011 at 8:23
• I learned a similar trick when I was younger and into magic tricks, of course with that one the only real trick is how long you're willing to practice flipping a coin.... Aug 31, 2011 at 16:52
• An iteresting answer to a different question. Aug 31, 2011 at 21:40
• That's why good referees toss the coin first, hide the result, and then ask a team leader to choose a result. Mar 27, 2012 at 0:14