This is really very obvious, and no doubt in the literature, but somehow hasn't occurred to me until now. Suppose that a fair coin is tossed an infinite number times. Suppose, further, than in the first hundred tosses it lands heads about half the time. It's no mystery why it lands heads about half the time in the first hundred tosses: it's because the probability of heads is 1/2 (plus properties of the binomial distribution). But suppose frequentism is true. Then the reason the probability of heads is 1/2 is that the infinite sequence has a limiting proportion of heads of 1/2. Now consider these three statements:
- A: approximately half of the first 100 tosses are heads
- B: the limiting proportion of heads is 1/2
- C: the limiting proportion of heads starting with the 101st toss is 1/2.
There are some gaps in the argument--explanation is hyperintensional, for instance. But I think the argument has a lot of intuitive force.