## Where Are The Unfair Coins?

I had been reading Anand Sarwate’s essay “Randomized response, differential privacy, and the elusive biased coin”. It’s about the problem of how to get honest answers when the respondent might feel embarrassed to give an honest answer. And that’s interesting in its own right.

Along the way Sarwate mentioned the problem of finding a biased coin. In probability classes and probability problems we often call on the “fair coin” or “unbiased coin”. It’s a coin that, when tossed, comes up tails exactly half the time, and comes up heads the other half. An unfair coin, also called a biased coin, doesn’t do that. One side comes up, consistently, more often than half the time.

Both are beloved by probability instructors and textbook writers. It’s easy to get students to imagine flipping a coin, and there’s only two outcomes of a coin flip. So it’s easy to write, and solve, problems that teach how to calculate the probabilities of various events. Dice are almost as popular, but the average cube die has a whopping six possible outcomes. That can be a lot to deal with.

Between my title and Sarwate’s title you likely know where this is going. Someone (Andrew Gelman and Deborah Nolan) finally got to ask the question: are there even unfair coins? And the evidence seems to be that you really can’t bias a coin. It’s possible to throw a coin so that a desired side comes up more often than chance. But it’s not inherent to the coin, unless it’s a double-headed or double-tailed coin. I’d always casually assumed that biased coins were a thing, just like loaded dice were. Now I have to reconsider that. I’d also doubt this loaded-dice thing. But would dozens of charming lightly comic movies about Damon Runyonesque gamblers lie to me?