Coin Flipping. Is it really even?

Liam was conducting a bit of an experiment today. His premise was that he needed to know which side of the coin was facing "up" on my thumb when I flipped it as this would influence his choice of whether to call heads or tails. He said there was a "Strong" advantage to calling the same side that faced up. I stated that it was probably a "slight" advantage, probably closer to something like 51% (instead of the experimental probability of 50%). Here is my evidence to prove that.

Here is (tongue in cheek) the SHORT answer

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).

To go to the website this is all from (trust me, it's fascinating stuff): Click HERE

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.

And HERE are there amazing 7 tips (for winning) at choosing a coin. I would only go into this much depth if my life was at stake, but still...

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 facedown. 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.