Suppose that at locations ...,−3,−2,−1,0,1,2,3,... (in time or one spatial dimension) a relevantly similar independent fair coin is tossed. Let Ln be the event that we have heads at all locations k≤n. Let Rn be the event that we have heads at all locations k≥n. Let A≤B mean that event B is at least as likely as A, and suppose all our events Ln and Rn are comparable (i.e., A≤B or B≤A whenever A and B are among the events Ln and Rn). Write A<B when A≤B but not B≤A. Assume ≤ is transitive and reflexive. Then, intuitively, we also have:
Now here is a surprising consequence of the above assumptions. Say that n is a switch-over point provided that Ln≥Rn but Ln+1<Rn+1. When there is a switch-over point, it's unique (since for m≤n, we will have Lm≥Ln≥Rn≥Rm, and for m>n, we will have Lm≤Ln+1<Rn+1≤Rm). Then:
- Under the above assumptions, either (a) there is a switch-over point, or (b) Ln<Rm for all n and m, or (c) Rn<Lm for all n and m.
So something is wrong with the assumptions. Classical probability theory says that what's wrong are (1) and (2): in fact, all of the Ln and Rn events are equally likely, i.e., have probability 0. This seems a very plausible diagnosis to me.
Why does (3) hold? Well, suppose that (b) and (c) do not hold. Thus, Ln<Rm for some n and m. If m≥n, then we Lm≤Ln<Rm, and if m<n, then we have Ln<Rm<Rn. In either case, there is an b such that Lb<Rb. By the same reasoning, by the falsity of (c), there is an a such that La>Ra. As n moves from a to b, then, Ln decreases while Rn increases, and we start with Ln bigger than Rn at n=a and end with Ln smaller than Rn at n=b. This guarantees the existence of a switch-over point.
This result is basically a generalization of an observation about Popper functions for such infinite sequences in a forthcoming paper of mine.