# 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?

## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

## 16 thoughts on “Where Are The Unfair Coins?”

1. Certainly, yes. A coin toss can be biased, but it’s not something that’s inherent to the coin.

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1. John Friedrich says:

A hemispherical coin would be biased. Imagine a quarter with a REALLY huge Washington’s head and the traditional tail.

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1. Presumably it would, sure. But at that point we are starting to challenge the idea of a coin.

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2. Since coin is non-uniform structure, and one who is flipping can give it certain spin or no-spin to get a biased result. My brother was a master of ‘toss’ and rarely lost when he was flipping or whenever he saw the ‘side of coin’ before flipping by other person. I felt he counted possible flips in air by same contender (usually me) everyday.

Good point! Loved ur post.

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1. I’m glad you liked it. And, yes, apparently it’s all in the toss to make an unfair coin toss. The coin itself is blameless in the outcome.

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3. I am a far cry from a mathematician, but the biased response is very interesting to me. Whenever I’m given a survey to fill out, I try to answer as honestly as possible….but if I’m in a hurry I don’t pay as much attention to the questions. But what if the person is embarrassed? Or purposely messes it up, as in someone who can have some control of a coin toss? Does that skew, in the case of a survey, the statistics beyond the built in margin of error?

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1. These things can screw up results, yes. People want to make themselves look better than they really are, or want to avoid admitting something embarrassing, or obfuscate saying they did something illicit or cruel.

The advantage of using a coin toss is that it allows cover. If I ask you to check ‘A’ if either you’ve used an illicit drug or a coin tossed comes up tails, then, your ‘A’ doesn’t really admit to anything. There’s no way to connect your answer to which meaning it could have. So, at least the reasoning goes, people shouldn’t feel inclined to lie or misrepresent themselves.

But if you have, say, 2,000 responses, and 1,100 ‘yes’ responses, then you can draw some conclusions. Pretty close to 1,000 of those ‘yes’ responses will be the coin toss coming up tails. So about 100 of the 1,000 remaining are ‘yes’ to the actual survey question.

But this does suppose that people aren’t screwing up the coin toss. On the other hand, since the point of a coin-toss-survey is to allow people to answer honestly under the cover of a coin toss’s result, why would someone do that?

No answer, really. People are perverse. But the people who’d fake a coin toss so they could falsely answer a survey question are, probably, fairly few, something within the margin of error for any survey. We would hope.

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1. Very good explanation. Maybe I shouldn’t be quite so negative about surveys. My one statistics class years ago taught me that any statistic can be looked at more than one way.

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1. Surveys and statistics are tools, and they’re quite powerful ones. They can help us make sure our judgement is sound. But they can’t do it all themselves; they’re just tools. People must make decisions.

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2. Geoffrey says:

Sorry I missed this thread at the time!

I work in statistics for a living, and yes, those issues are TREMENDOUSLY important in designing or interpreting a survey. My work is more on the mathematical side of things, working out stuff like the error we get from “sampling effects” (i.e. how far can we generalise from the people we surveyed to the people we didn’t survey?) but we also have people with psych-type training who pay a lot of attention to that sort of non-sampling stuff.

One of the surveys we run is an expenditure survey, where we ask people what they’re spending their money on (important for calculating things like inflation stats). If you send out a respectable-looking middle-aged lady to ask somebody how much they spend on smoking, drinking, and gambling, the answers will be VERY different from the ones they’d give to a twenty-year-old guy.

Another one where embarrassment causes big difficulties – if you ask people directly “how often do you go to church?” you’ll hear that your respondents are really pious folk. But if you ask them to fill out a diary that records what they do with their time, rates for church attendance are much much lower. Often we have to get a little sneaky in order to get data that isn’t skewed by that “social desirability bias”. Sometimes web-based surveys can help; there are times when it’s useful to have an interviewer who can help people understand the questions, but anonymity might give more honesty.

When you see a margin of error reported for statistics, usually it’s only for the sampling error; it’s very hard to estimate the non-sampling error. (You can try using two different methods and comparing the results, but that’s expensive and still doesn’t tell you what the true answer is.) Unfortunately a lot of folk take the attitude that if they can’t see it, it doesn’t matter… definitely NOT true.

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1. I appreciate having the professional review of this. Thank you. I’m aware of the problems of designing a good survey but only from reading in statistics and psychology texts about the problems. I’ve been fortunate not to face them in my day jobs yet.

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