Vincent Gable’s Blog

Flipping a coin is, ever so slightly, unfair. As this article (via) points out, there is a bias for a coin to land on the same side it started on.

Fortunately, all the biases coins have are systemic biases — they effect all similar coins the same way.

So, with a fair thrower, it’s possible to flip twice, and have the bias of the two throws cancel each other out.

Procedure

1. Put a coin heads-up, and flip it, as you normally would.
2. Note the result, if certified this will be the decision.
3. Flip the coin again, exactly as you did in step 1.
4. If the coin lands on the opposite side as it did in step 2, the result from step 2 is certified. Otherwise, restart from step 1.

For maximum fairness and reproducibility, it’s best to let the coin land on the floor.

Why This Works

To simplify discussion, let’s call the sides of the coin unlikely (U) and likely (L) instead of heads & tails.

There are only 4 possible results to a pair of coin tosses: UU, UL, LU, LL. Obviously LL is most likely, and UU is least likely, so we rethrow if we get either (steps 3-4). That means the only “certified” results are UL or LU, and the odds of getting UL are the same as getting LU.

Dexterous Cheating

Unfortunately, this is not a self-enforcing protocol, so if the thrower is skillful enough, they can make the second throw go however they like, and keep re-throwing until they get the result they want.

Fortunately, most people aren’t able to manipulate a coin-toss. If you are worried that someone else is, then only let them flip once, and call the result in the air — that way they won’t know which side to pick.

If you can throw the result, and can’t find someone else to call the result — it serves you right for driving away all your friends by cheating at coin tosses, you tosser. But I’m still impressed.