Monday, April 14, 2014

A curious thing about infinite sequences of coin tosses

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 kn. Let Rn be the event that we have heads at all locations kn. Let AB mean that event B is at least as likely as A, and suppose all our events Ln and Rn are comparable (i.e., AB or BA whenever A and B are among the events Ln and Rn). Write A<B when AB but not BA. Assume ≤ is transitive and reflexive. Then, intuitively, we also have:

  1. Ln+1<Ln
  2. Rn<Rn+1.
After all, Ln+1 and Rn require one more heads result than Ln and Rn+1, respectively.

Now here is a surprising consequence of the above assumptions. Say that n is a switch-over point provided that LnRn but Ln+1<Rn+1. When there is a switch-over point, it's unique (since for mn, we will have LmLnRnRm, and for m>n, we will have LmLn+1<Rn+1Rm). Then:

  1. 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.
But each of these options is really rather absurd. Option (a) says that the probability distribution of our sequence of relevantly similar independent fair coins has a distinguished switch-over point. Option (b) implies that infinite sequences of heads stretching leftward (if we're talking about a spatial arrangement; backward, if temporal) are always less likely than infinite sequences of heads stretching rightward (forward, if temporal). And option (c) is just as absurd as (b). And of course in each of the three cases we have a violation of very plausible symmetry conditions on the story: in case (a), we have a violation of symmetry under shifts, and cases (b) and (c) we have a violation of symmetry under flips.

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 mn, then we LmLn<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.

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