Statistics, Coin Tosses, and Expected Values

In class, I often illustrate the concept of expecvcted value with the example of a game where a coin is tossed - if it comes up heads, it pays a $2, if it comes up tails, it pays $1. If the coin is a "fair" one, the expected value of the coin toss is $1.50. If the coin is not a fair one (i.e. if the probability of a head is, say, 75%), the expected value is not $1.50. In the case where the probability of a head is 75%, the expected value is (0.75 x 2.00) + (0.25 x 1) = $1.75.

Now I find out (according to Andrew Gelman at Statistical Modeling, Causal Inference, and Social Science) that a coin can't be biased - the probability of a flipped coin coming up heads MUST be 50%:

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.

Dang! I guess I'll have to get a new analogy.