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The unicorn of probability theory

“A coin with probability p > 0 of turning up heads is tossed . . . ” — Woodroofe, Probability with Applications (1975, p. 108)

“Suppose a coin having probability 0.7 of coming up heads
is tossed . . . ” — Ross, Introduction to Probability Models (2000, p. 82)

The biased coin is the unicorn of probability theory—-everybody has heard of it, but it has never been spotted in the flesh. As with the unicorn, you probably have some idea of what the biased coin looks like—-perhaps it is slightly lumpy, with a highly nonuniform distribution of weight. In fact, the biased coin does not exist, at least as far as flipping goes.

You can load a die, but you can’t bias a coin

You can load a die (that is, weight it or bevel it so that, when rolled, all 6 sides are not equally likely to land face-up), but you can’t bias a coin (that is, weight it or bevel it so that, when flipped and caught in the air, it is more likely to land “heads” than “tails” (or vice-versa). The difference is: the die bounces and so weighting and beveling can affect how it lands; the coin is just flipping in the air (with little air resistance), and there is no way to make either “heads” or “tails” spend more than half the time facing up in its spin.

Deborah Nolan and I discuss the issue further (and also give a fun classroom demo) in this article, and also in this book.

But . . . commenters Scott and Marianolake on this blog pointed me to a recent study found that coins, when flipped, “will land the same way it started about 51 percent of the time.”

No, you can’t build a biased coin (if it’s going to be flipped in the air)

Does this contradict our finding that “you can’t bias a coin”? No. As I wrote in my reply to Scott, the cited article says that there is a 51% chance the coin will land “the same way it’s started,” not a 51% chance of heads. Depending on the method of flipping, the probability of landing on the same side as started can be varied. But, if the coin is flipped high enough and with enough uncertainty that the initial state is “forgotten,” then the probability of “heads” will necessarily be 50%.

This is assuming the coin is flipped and caught in the air–not spun, not allowed to bounce.


  1. MDM says:

    Actually, I think you're wrong. Consider a coin whose center of mass is close to the rim, and is flipped such that its axis of rotation is perpendicular to the line formed by the center of mass and the radius. I think that the momentum of the spin would carry it around so that, on landing, it would land on one side for more than 180 degrees of its rotation. That is, if the coin hits the (flat) hand on the weighted side, even if the angle is > 90 degrees, it would continue to rotate toward 0 degrees.

  2. Andrew says:

    Hmmm, maybe you're right! I'd still stand by my claim that that Ross's coin wiht Pr(heads)=0.7 doesn't exist anywhere. Continuing the unicorn analogy: it doesn't exist, but maybe it could be built…

  3. Mike Anderson says:

    So flip something non-homogeneous, like thumbtacks.

  4. Mikhail says:

    What about throwing knives? Properly balanced – i.e., handle-vs.-blade – they stick blade-first into their target even in the hands of an amateur. The knife is never allowed to bounce, so it is possble to bevel something so that its rotation through the air is affected.

  5. Andrew says:


    I am certainly no expert on knife-throwing (!) but my guess is that what's going on here is that there are so few spins that a thrower can control it. Also, when throwing a knife, the thrower always starts by grabbing the handle, not the blade (or so I assume), and so he or she just has to toss it so that Pr(initial side lands first) is low. With coin flipping, the initial state is arbitrary, which is why there's no way to have Pr(heads)=0.7 (even if you can cleverly toss so that Pr(initial side lands first)=0.7, for example).

  6. Maribel says:

    I've seen an unicorn, or I think so. The Spanish 1 euro coin is biased. When you flip one of this coins, heads seem to have a bigger probability than tails. In fact, football referees don't use them as they are unbalanced!!!

  7. Andrew says:


    I'm skeptical. Try flipping the coin 100 times.

  8. pm says:

    I'm a 7th grader doing a science project. I've watched people checking out at the supermarket. The lines in the center always have the most people. My theory was that people would choose the shortest line and it would be random. This is not the case. Anyway I was wondering if you guys knew of any probability theory to predict what lines a group of people would choose.

    thank you

  9. Joel says:

    I was trying to find some info online and came up with nothing. Perhaps someone here can help. What I'm trying to find out is whether or not any common US coin (particularly a quarter) has an uneven weight distribution on either side. To state it differently: do the designs protruding from the metal on the heads side have the same weight as the those on the tails side? If they aren't exactly equal (and the coin is allowed to bounce off any surface until it lies still – as most people tend to do), then knowing which side weighs slightly more can give you a small advantage in determining the outcome (however miniscule it may be).

  10. Kimberlee Gerritzen says:

    Have any of you all done an experiment on whether or not you could predict the outcome when flipping a coin(a quarter)?

  11. qwertyuiop says:

    I disagree with MDM. any alteration to the centre of mass will be at such a small angle from the rim, it will effectively have no effect upon the rotation. And as you said, to have a chance of making a difference, you would have to spin it with incredible skill. A 10p coin, for instance, has a diameter of 24.5mm and a thickness of 1.5mm. therefore the angle you span it at would have to be arcsin 1.5/24.5 = 3.5 degrees. it is not possible to spin a coin sideways without effectively throwing it on the floor

  12. Zach Dwiel says:

    What about a buttered piece of bread that always seems to land butter-side down? Could one side of the 'coin' be made with a different texture such that the air travels slower than the other creating uneven pressure and a small but noticeable effect on which side lands face down? Maybe try flipping a piece adhesive velcro ?

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