The Coin Flip: A Fundamentally Unfair Proposition?

Have you ever flipped a coin as a way of deciding something with another person? The answer is probably yes. And you probably did so assuming you were getting a fair deal, because, as everybody knows, a coin is equally likely to show heads or tails after a single flip—unless it's been shaved or weighted or has a week-old smear of coffee on its underbelly.



So when your friend places a coin on his thumb and says "call it in the air", you realize that it doesn't really matter whether you pick heads or tails. Every person has a preference, of course—heads or tails might feel "luckier" to you—but logically the chances are equal.

Or are they?

Granted, everybody knows that newly-minted coins are born with tiny imperfections, minute deviations introduced by the fabrication process. Everybody knows that, over time, a coin will wear and tear, picking up scratches, dings, dents, bacteria, and finger-grease. And everybody knows that these imperfections can affect the physics of the coin flip, biasing the results by some infinitesimal amount which in practice we ignore.

But let's assume that's not the case.

Let's assume the coin is fabricated perfectly, down to the last vigintillionth of a yoctometer. And, since it's possible to train one's thumb to flip a coin such that it comes up heads or tails a huge percentage of the time, let's assume the person flipping the coin isn't a magician or a prestidigitator. In other words, let's assume both a perfect coin and an honest toss, such as the kind you might make with a friend to decide who pays for lunch.

In that case there's an absolute right and wrong answer to the age-old question...

Heads or tails?

...because the two outcomes of a typical coin flip are not equally likely.

The 50-50 proposition is actually more of a 51-49 proposition, if not worse. The sacred coin flip exhibits (at minimum) a whopping 1% bias, and possibly much more. 1% may not sound like a lot, but it's more than the typical casino edge in a game of blackjack or slots. What's more, you can take advantage of this little-known fact to give yourself an edge in all future coin-flip battles.

The Physics of Coin Flipping

In the 31-page Dynamical Bias in the Coin Toss, Persi Diaconis, Susan Holmes, and Richard Montgomery lay out the theory and practice of coin-flipping to a degree that's just, well, downright intimidating.

Suffice to say their approach involved a lot of physics, a lot of math, motion-capture cameras, random experimentation, and an automated "coin-flipper" device capable of flipping a coin and producing Heads 100% of the time.



Here are the broad strokes of their research:

1. If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there's a 51% chance it will end as heads).
2. If the coin is spun, rather than tossed, it can have a much-larger-than-50% chance of ending with the heavier side down. Spun coins can exhibit "huge bias" (some spun coins will fall tails-up 80% of the time).
3. If the coin is tossed and allowed to clatter to the floor, this probably adds randomness.
4. If the coin is tossed and allowed to clatter to the floor where it spins, as will sometimes happen, the above spinning bias probably comes into play.
5. A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity.
6. The same initial coin-flipping conditions produce the same coin flip result. That is, there's a certain amount of determinism to the coin flip.
7. A more robust coin toss (more revolutions) decreases the bias.

The 51% figure in Premise 1 is a bit curious and, when I first saw it, I assumed it was a minor bias introduced by the fact that the "heads" side of the coin has more decoration than the "tails" side, making it heavier. But it turns out that this sort of imbalance has virtually no effect unless you spin the coin on its edge, in which case you'll see a huge bias. The reason a typical coin toss is 51-49 and not 50-50 has nothing to do with the asymmetry of the coin and everything to do with the aggregate amount of time the coin spends in each state, as it flips through space.

A good way of thinking about this is by looking at the ratio of odd numbers to even numbers when you start counting from 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

No matter how long you count, you'll find that at any given point, one of two things will be true:

• You've touched more odd numbers than even numbers
• You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

• You've touched more even numbers than odd numbers.

Similarly, consider a coin, launched in the "heads" position, flipping heads over tails through the ether:

H T H T H T H T H T H T H T H T H T H T H T H T H

At any given point in time, either the coin will have spent equal time in the Heads and Tails states, or it will have spent more time in the Heads state. In the aggregate, it's slightly more likely that the coin shows Heads at a given point in time—including whatever time the coin is caught. And vice-versa if you start the coin-flip from the Tails position.

The Strategy of Coin Flipping

Unlike the article on the edge in a game of blackjack mentioned previously, I've never seen a description of "coin flipping strategy" which takes the above science into count. When it's a true 50-50 toss, there is no strategy. But if we take it as granted, or at least possible, that a coin flip does indeed exhibit a 1% or more bias, then the following rules of thumb might apply.

1. Always be the chooser, if possible. This allows you to leverage Premise 1 or Premise 2 for those extra percentage points.
2. Always be the tosser, if you can. This protects you from virtuoso coin-flippers who are able to leverage Premise 6 to produce a desired outcome. It also protects you against the added randomness (read: fairness) introduced by flippers who will occasionally, without rhyme or reason, invert the coin in their palm before revealing. Tricksy Hobbitses.
3. Don't allow the same person to both toss and choose. Unless, of course, that person is you.
4. If the coin is being tossed, and you're the chooser, always choose the side that's initially face down. According to Premise 1, you'd always choose the side that's initially face up, but most people, upon flipping a coin, will invert it into their other palm before revealing. Hence, you choose the opposite side, but you get the same 1% advantage. Of course, if you happen to know that a particular flipper doesn't do this, use your better judgment.
5. If you are the tosser but not the chooser, sometimes invert the coin into your other palm after catching, and sometimes don't. This protects you against people who follow Rule 4 blindly by assuming you'll either invert the coin or you won't.
6. If the coin is being spun rather than tossed, always choose whichever side is lightest. On a typical coin, the "Heads" side of the coin will have more "stuff" engraved on it, causing Tails to show up more frequently than it should. Choosing Tails in this situation is usually the power play.
7. Never under any circumstances agree to a coin spin if you're not the chooser. This opens you up to a devastating attack if your opponent is aware of Premise 2.

I hope I've made it clear that none of this is really to be taken seriously. The point is that adding even 1% of wobble to a situation of pure chance can create a lot of additional complexity, and that in turn, can create strategy where none existed before.