## Flip Best

I’ve just learned a little trick to get a fair toss with an unfair coin. Suppose you have a coin that lands heads 51% of the time. Flipping it to make a decision is unfair (even if just a little). To fix this and make a fair random toss, flip the coin twice. If the tosses are the same, redo it. If they’re different, use the second toss as the outcome. For example, if you flip HH, ignore it and flip again. If you then flip HT, tails wins. This works because HT and TH are equally likely. It works no matter how biased the coin is (except 100%), but with a very biased coin you may have to try several times before getting a HT or TH sequence.

This procedure adds some additional suspense to the decision-making process. With one flip it’s over and done in a blush. With two (or more) flips, after the first flip, you still don’t know the outcome, but the probabilities of heads or tails winning have changed from 50%. Whichever side comes up first is now more likely to lose because the opposite side is just one flip away from winning. You might have to go back and reset everything, though. If you wind up having to toss out three flip-pairs in a row before making the decision, it would be an intense coin flip.

We could imagine a different coin that’s also unfair. It has a 51% chance to land whichever way it was flipped. Heads and tails are overall equally likely, though. We could call the first coin “side-biased” and the second one “change-biased”.

Can we get a fair toss with the change-biased coin in the same way we got one with the side-biased coin?

Yes. If the coin starts out with heads facing up, then the equally-likely sequences of two flips are HT and TT. The first flip is the winner, but you don’t know if it’s won until the second flip. This works out beautifully because you have to flip again if you get HH or TH. That’s good since it means you start the second double flip on heads – the same as the first.

In retrospect, it’s obvious that the strategy from the side-biased coin carries over to a strategy for the change-biased coin because the two coins can emulate each other.

Imagine you don’t have a biased coin at all, so you call up your friend and ask her to do some flips for you with a side-biased coin. She gives you a sequence of flips seems pretty much random, but has heads coming up more often than tails. So you conclude she has a side-biased coin, since a change-biased coin would give heads and tails equally in the long run.

Actually, though, your friend has a change-biased coin. She’s just tricking you. If she gets no change, she says “heads” aloud, regardless of which side is actually up. If she gets a change, she says “tails”. This way she creates independent, biased heads/tails data, the same as if she truly had a side-bias coin.

You figure out the trick, go over to her house, and secretly replace her change bias-coin with a side-bias coin you made just for the purpose. The next day you call her up again, this time asking for the results of flipping a change-bias coin. You think you’ll trick her, because she doesn’t even have one. Once she sees heads are coming up more frequently, she’ll realize her change-bias coin has been switched, and the joke will be on her.

To your astonishment, she gives you a long sequence of tosses that are 50% heads and 50% tails, but has the “streakiness” characteristic of a change-bias coin! What happened is that she saw you make the switch, investigated the new coin, and realized it was a side-biased coin. Then she took a piece of paper, wrote “heads” on one side and “tails” on the other. When you asked her for change-biased flips, she flipped her side-biased coin. If it came up heads, she flipped her piece of paper over and read out what was on it. If her coin flip came up tails, she read what was on the paper without flipping it. In this way she mimicked the process of a coin deciding whether to change or not with a bias.

Not only can the side-biased and change-biased coins both be made fair, either one can do anything the other can, because it can perfectly well pretend to be the other one.