You are one of infinitely many blindfolded people arranged in a line with a beginning and no end. Some people have a red hat and others have a white hat. The process by which hat colors were assigned took no account of the order of people. You don't know where you were in the line. Suppose you learn the exact sequence of hat colors, say, RWRRRRWRWRWWWWRWWWR.... But you still don't know your position. What should your probability be that your hat is red?
A natural way to answer this is to compute the limiting frequency of reds. Let R(n) be the number of red hats among the first n people, and then see if R(n)/n converges to some number. If so, then that number, call it r, seems to be a reasonable value for the probability. Call the assignment of r to the probability when the limit r exists the frequency rule.
Here's a curious and simple thing I hadn't noticed before. If you think the frequency rule is always the right rule, then for all integers n, you are committed to being almost certain that your position is greater than n. Here's why. Suppose that the sequence that comes up is n white hats followed by just red hats. The limiting frequency of R(n)/n is 1. So by the frequency rule, you're committed to assigning probability 1 to having a red hat. But since you have a red hat if and only if your position is greater than n, you are committed to assigning probability 1 to your position being greater than n. And since there is no connection between the hat color arrangement and the order of people on the line, if you have this commitment after learning the sequence of hat colors, you also had it before. The argument applies for all n, so for all n you must have been almost certain that your position in the sequence is greater than n.
And this in turn leads to the paradoxes of nonconglomerability. For instance, suppose that I flip a fair coin. If it's heads, I let N be your position number. If it's tails, I choose a number N at random such that P(N=n)=2−n. In either case, I reveal to you the value of N, but not how the flip went. For any number n, the probability that N=n is zero given heads (since you're almost certain that your position is greater than n), and the probability that N=n is greater than zero given tails, so by Bayes' Theorem you will be almost certain that the coin landed tails. So I can make you be sure that a coin landed tails, and thereby exploit you in paradoxical ways.
So the frequency rule isn't as innocent as it seems. It commits one to something like an infinite fair lottery.