No Biased Coins

Many introductory probability courses involve 'biased' coins. The authors of this paper argue this is the unicorn of probability theory: everyone has heard of it, but no one has seen it, because it doesn't exist!

Dice can be loaded—that is, one can easily alter a die so that the probabilities of landing on the six sides are dramatically unequal. However, it is not possible to bias a coin flip—that is, one cannot, for example, weight a coin so that it is substantially more likely to land “heads” than “tails” when flipped and caught in the hand in the usual manner. Coin tosses can be biased only if the coin is allowed to bounce or be spun rather than simply flipped in the air