Monday, August 24, 2009

Coin flipping is less random than commonly believed:
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.
Here are the strategy implications:
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.
Here's the academic paper. (Via Bruce Schneier.)