Will Knott

A random coin toss with an evenly weighted coin is one of the basics of statistics. Toss a coin and is a 50-50 chance of coming up heads. Or tails. Or harps or whatever you are supposed to call on the Euro.

Yes, I have had a “heads of tails” toss be questioned with “which side is heads” while the coin is in the air more than once, and no one able to answer. Danged problem with a maps on one side and a whole differing bunch of symbols on offer on the other.

Anyway, 50-50, right?

Actually, no. It is a 49-51 percent chance.

Why? Logic, and physics. It turns out that the problem is a statistical bias as a result of dynamical bias.

Lets start with a coin heads (H), and toss it. As it moves through the air, the upward facing side changes.

H-T-H-T-H-T-H-T.. and so on.

However, if you start with heads facing up you have two scenarios.

1. You have more heads than tails.
2. You have an equal number of heads and tails.

At no point will you have

• more tails than heads

as the starting conditions demand it can’t be so.

It simply can’t happen. Laws of physics.

Of course, this is a 1% chance. A casino has less than a 1% house bias for many of its games.

The funny things is, if the coin is spun (think about the cliche of the bored gangster spinning a coin with his thumb and catching it in mid-air… or Two-Face actually), and left drop too the floor, the percentages changes.

The odds of the heavier side hitting the ground can be up to 80%!

It depends a lot of the coin naturally, but its a big jump.

The heavier side tends to be the more detailed side, which means that spinning a lot of Euro coins should result in a lot of maps facing up from the ground.

Don’t take my word for it, read the “Dynamical Bias in the Coin Toss“, a 2007 paper by Persi Diaconis, Susan Holmes and Richard Montgomery. Personally I love the low tech method of sticking a ribbon to a coin, tossing it and counting the number of twists in the ribbon, but they used high speed photography too. The paper ends with the following useful tips.

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. It happens
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.

I will admit that it sounds worthy of an Ig Noble Award, but for one thing. It shows that and why a long held logical assumption is in fact wrong. Meaning that it is actually pure science.

Flipping marvelous (sorry),

Will