Not So Random After All

The random coin toss must be one of society’s most frequently used decision-making mechanisms. We use the coin toss to choose which movie to see, to determine team positions in major sporting events, to divvy up household chores, and even name cities. But it may be that the the random coin toss isn’t so random. A 2007 study found that a vigorously flipped coin is likely to land on the same side it started on at least 51 percent of the time, possibly more depending on the person doing the flipping. (HT: Chris Blattman)[%comments]


They don't just flip coins at most games because everyone knows the flipper can control the outcome. They let the coin fall and bounce. They usually now call it a coin toss for that reason.

Walter M Lamia

With some practice, a good slight-of-hand magician can flip a coin and consistently get whatever face s/he wants.

Different weight on each side because of the engraving is also a source of bias.

Coin flipping is notoriously manipulable, and is a terrible randomizing device. Makes you wonder why the NFL depends on it for overtime games, since the first team to get the ball has a much higher rate of winning.


Refer to papers by Diaconis and Engel as well as chapter 10 in Jaynes ("Physics of 'random experiments'" Probability theory: the logic of science)


Manipulation by the flipper shouldn't be an issue with the flipper is an independent third party (like a referee). And letting it bounce off the ground probably helps too.


Isn't any advantage negated by having one person flip, and the other person call it in the air? This would be true even if the flipper had 100% control over which side landed up, because he wouldn't know which side his opponent would choose until after he had flipped it. (Effectively, this would just turn the coin toss into the game "odds and evens.")


come on Walt, if the coin flip was really that unfair wouldnt all teams pick the higher outcome everytime? and which one IS the unfair outcome? In a game of inches, where the home team is already at a significant advantage and the outcome of one pass tipped in the air adds far more variance to the outcome than the coin flip I dont think it is a problem at all in the NFL. I am more concerned about the silly roughing the passer rules!

It would be fun to see the success rate of the visiting team correctly calling the toss through NFL history... my bet is that it is pretty close to 50.000%

the Gooch

Nothing beyond the realm of quantum physics is truly random.

Eric H.

"Makes you wonder why the NFL depends on it for overtime games, since the first team to get the ball has a much higher rate of winning."

Except, of course, they don't. It's about 52-48.

Howard Tayler

@Ari: My dad was able to win EVERY TIME if he flipped the coin and you called it in the air. How? By catching the coin and manipulating it as he slapped it onto the back of his other hand.

You can only negate this by letting the coin fall onto the ground.

Peter Bouman

I'm not sure I understand the title of this column.
What you say is that: "a vigorously flipped coin is likely to land on the same side it started on at least 51 percent of the time."
This statement doesn't imply that the process is not random; you're saying that the toss is not necessarily independent of the starting conditions, which is completely different.
The implication is that the probabilities of heads or tails have some dependence on the starting conditions.

"Not so random" should really be replaced by "Not so independent" or "Not so unbiased."

Kevin Canini

@Peter Bouman: You're right... using the statistical definition of "random", the biased coin toss is still random. However, the vast majority of the public doesn't understand the technical definition and uses "random" to mean something like "all outcomes have equal probability".


wow- this makes no country for old men that much more profound...


#2 Walter

Makes you wonder why the NFL depends on overtime to decide games at all, the NCAA system is much more equitable.


Persi Diaconis (the first author on the paper), the colorful coin-tossing, card-shuffling mathemagician has been studying this stuff for decades. My Stanford roommate had him a couple years back for stats; word on the street is he has trained himself to toss a heads every time. Made me wonder if he was in fact that missing 1%.

Ian Stanczyk

Random? Just ask Rosencrantz.


I read about a study not too long ago which sounds a lot like this study (I'm not sure if it's really the same one since I am having trouble loading the pdf linked here.)

In the study I refer to, the reason for the coin landing more often on the side it started was because every once in a while the coin failed to flip at all, even though it looked like it was flipping.

The way this happens is that the coin "wobbles" in much the way that a coin spun on a table would. This causes the coin to look like it is flipping when it is not. Some magicians who control coin flips may have simply learned to reproduce this behavior.

Most of the coin flips in the study that I read about were actually performed by a coin-flipping robot and the "wobble" effect was confirmed by watching slow-motion video of coin tosses.

While there are certainly people who can manipulate their own coin tosses (possibly by using reproducing the "wobble" behavior) I do not think that that this study's results were influenced by a rogue magician getting into the sample.

Concerning the method of having one person flip and the other call, this would only eliminate advantage if the caller couldn't see the initial up-side of the coin.



What would be interesting is the frequency distribution of the number of half-turns of the coin in the air. A sufficiently wide distribution would normally give you close to 50-50.

The only way an odd or even number of half-revolutions would be highly likely is if a particular number of turns was very frequent.


personally I don't quite like the coin flipping technique... what I prefer to do is... hold the coin in the hand and ask the other person to guess if its head or tails... the chances of this being biased are basically NIL...


Coin tossing? Pffft. Anyone who doesn't use Rock-Paper-Scissors as their deciding mechanism is just foolish.


hello... one side of the coin is usually heavier then the other, so it's center of balance is not at it's actual center. if you had a flat disk with a black ink marking on one side and a red ink marking on the other, you might get closer to 50%.