Arnold Kling ponders probability:
How one thinks about probability affects how one thinks about economics. Consider the use of the word "probability" in each of the following sentences:
1. What is the probability that when a fair coin is flipped it will come up heads?
2. What is the probability that exactly two number-one seeds will make it to the final four in the March Madness basketball tournament?
3. What is the probability that New York City will rank higher relative to other cities five years from now in terms of college graduates?
We would answer the first question by saying that the probability is 50 percent, based on the very definition of a fair coin. This is an axiomatic interpretation of probability. The axiomatic view treats probability as a matter of pure logic, with statements that do not require any empirical testing.
We would answer the second question by looking up historical records for the NCAA basketball tournament. This is the frequentist account of probability, which treats probability as counting outcomes from repeated trials. A frequentist would claim that the only way we can know that a coin has a 50 percent probability of coming up heads is by actually flipping a coin enough times to verify this empirically.
The third question cannot be answered on the basis of axioms or observed frequencies. The probability estimate is purely subjective. The subjective account of probability is that it reflects an individual belief that cannot be proven either logically or empirically.
Edit: Eliezer has in the past been uncomplimentary of the "accursèd frequentists". In at least Beautiful Probability and Probability is in the Mind, he has characterized (for at least some problems) the "frequentist" approach as being wrong, and the "Bayesian" approach as being right. Kling suggests different problems for which different approaches are approrpriate.