Getting a Fair Toss From a Biased Coin

A paper out of Stanford and UCSC, Dynamical Bias in the Coin Toss (PDF, 5MB), reveals a small but significant bias in the process of flipping a coin.

James Devlin of Coding the Wheel provides the following summary:

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 article goes on to suggest a coin-flipping strategy that tries to take advantage of the small bias, but what about getting a completely fair toss from a biased coin?

It turns out to be fairly easy. Just follow these steps:

1. Flip the coin twice.
2. If both tosses are the same (heads-heads or tails-tails), repeat step 1.
3. If the tosses come up heads-tails, count the toss as heads. If the tosses come up tails-heads, count it as tails.

To see why this method makes even a biased coin fair, let's pretend we have a weighted coin that comes up heads 60% of the time. If you toss it twice and throw out the result when both tosses are the same, you're left with two possible outcomes. The probabilities of the two remaining outcomes are the same.

P(HT) = P(TH)
P(H) * P(T) = P(T) * P(H)
0.6 * 0.4 = 0.6 * 0.4 = 0.24

Since both outcomes have exactly the same probability, the bias is removed. This method will work no matter how biased the coin you use, as long as there's some possibility of it coming up either heads or tails (so no two-headed coins allowed).

Note: This method for simulating a fair coin using a biased coin was introduced by John von Neumann in Various techniques used in connection with random digits (1951).