Probability and Infinite Coin Tosses
Have been reading an article by Timothy Williamson that I find quite interesting:
http://www.philosophy.ox.ac.uk/faculty/members/docs/infiniteheads.pdf
Suppose that a fair coin is tossed infinitely many times at one second intervals starting from t1, and the tosses are all independent. Let (H1…) be the event that every toss of the coin lands heads. Let (H1) be the event that the first toss at t1 lands heads, and (H2…) be the event that every toss from t2 (separated from t1 by one second) onwards lands heads.
There’s an intuition that (H1…), although extremely unlikely, is possible, and hence, we should not be certain that ~(H1…) - we should be more confident that (H1…) than we are that A & ~A. Williamson shares the intuition, but argues that with a few other plausible assumptions, it leads to contradiction.
Let “>” be read as “is more probable than” and “≥” be read as “is as least as probable as”. The following, Williamson thinks, is a “promising principle” for comparative probability:
(!) If X and Y are each incompatible with Z, then
(a) X > Y iff X v Z > Y v Z
(b) X ≥ Y iff X v Z ≥ Y v Z
But now, let X be (H1…), Y be A&~A, and Z be ~(H1) & (H2…). Since (H1…) and A&~A are both incompatible with ~(H1) & (H2…), we get:
(H1…) > A&~A iff (H1…) v (~H(1) & (H2…)) > A&~A v (~H(1) & (H2…))
(H1…) v (~H(1) & (H2…)) is equivalent to (H2…), and A&~A v (~H(1) & (H2…)) is equivalent to (~H(1) & (H2…)). Hence we get:
(H1…) > A&~A iff (H2…) > (~H(1) & (H2…))
But ~H(1) & (H2…) and (H1…) differ only on the outcome of the first toss; hence,
(H1…) > A&~A iff (H2…) > (H1…)
Now Williamson thinks that (H1…) ≥ (H2…), which means, given the equivalence above, that it’s not the case that (H1…) > A&~A. But this contradicts what intuition purportedly tells us. What is Williamson’s reason for thinking that (H1…) ≥ (H2…)? He writes:
“H(1…) and H(2…) are isomorphic events. More precisely, we can map the constituent single-toss events of (H1…) one-one onto the constituent single-toss events of H(2…) in a natural way that preserves the physical structure of the set-up just by mapping each toss to its successor. H(1…) and H(2…) are events of exactly the same qualitative type; they differ only in the inconsequential respect that H(2…) starts one second after H(1…). That H(2…) is preceded by another toss is irrelevant, given the independence of the tosses. Thus H(1…) and H(2…) should have the same probability.”
I have to admit that my mathematically naive intuition is that (H2…) > (H1…). Suppose that this is not correct, that (H1…) ≥ (H2…). Since (H1) & (H2…) ≥ (H1…), transitivity yields (H1) & (H2…) ≥ (H2…). But it seems to me that (H1) & (H2…) < (H2...). Imagine there's an evil demon who can choose to exercise his ability to tell, before we start to toss a coin infinitely, whether it'll land heads on every toss or not. You've previously been condemned to be tortured for eternity, but the evil demon offers you a way out. He gives you two tickets, one which says (H2...) and the other which says (H1). The coin has yet to be tossed, and the demon, who has thus far refrained from exercising his ability, tells you that you'll be spared from eternal torture if every ticket you hold says something true. Before the demon exercises his unique ability, he gives you the choice to throw away the ticket which says (H1) if you so desire. Should you throw it away?
It seems to me that you should. If (H1) is true, throwing the ticket away does not make it less likely that you'll be spared. If (H1) is false, then throwing it away is better than keeping it, since keeping it means that there's an additional possibility, and hence, additional danger, that you won't be spared. But this means that keeping the ticket decreases the probability that you'll be spared, i.e., (H1) & (H2...) < (H2...).
For all I know, defending the intuition that (H2...) > (H1…) might be fraught with insuperable mathematical problems. In any case, I find the intuition about as strong as the intuition that (H1…) > A&~A. If, by keeping the ticket which says (H1), the additional possibility that I won’t be spared doesn’t mean a lower probability that I’ll be spared, then it’s hard to see why the mere possibility that (H1…) means that I shouldn’t be certain that ~(H1). If it does not make any difference whether I keep or throw away the ticket saying (H1), then I’m inclined to think that a ticket saying (H1…) is worth exactly the same as one saying A&~A.
August 16th, 2007 at 7:43 pm
Maybe I’m missing something here (I confess I haven’t read Williams’s paper yet) but isn’t the probability of Hn zero? I mean, the probability of H1 is the limit of 0.5^n as n-> infinity, which is zero. Exactly the same applies to the others, since infinity - n is still infinity. This coincides with a reasonable intuition that all the Hn are impossible. Hence I can throw away neither, either or both tickets; I’m doomed either way.
August 20th, 2007 at 4:06 pm
Hi Ornette,
Some people have the intuition that (H1…), although extremely unlikely, is possible. And because it is possible, they think we shouldn’t be certain that ~(H1…), even if standard probability theory says that P((H1…)) = 0. In his paper, Williamson considers two ways to preserve the intuition. The first appeals to infinitesimals - on this approach, P((H1…)) is infinitely small, but is not zero. The second abandons quantitative probabilities in favour of qualitative probabilities, and this supposedly allows one to maintain that (H1…) is more probable than an outright contradiction. Williamson argues that both approaches lead to problems of their own. In my post, I focused on Williamson’s treatment of the second approach. It seems to me that the intuition that we should not be certain that ~(H1…), and which motivates the second approach that WIlliamson considers, is similar to the one that Williamson has to reject for his argument to work, namely, the intuition that (H2…) > (H1…). So either we reject both intuitions, and all is fine (we can retain standard probability theory), or we accept both intuitions, in which case pace Williamson, the second approach to dealing with the first intuition might work after all.
Hope that makes sense!
August 27th, 2007 at 1:25 am
Thanks Wenghong, that’s certainly useful. I usually find myself intuitively drawn to the ideas Williamson defends, but this is an odd one to me. People just don’t have good intuitions about infinities, because they’re not something we ever meet in everyday life, so I’m not sure we should really entertain whatever intuitions we might have…
Anyway, I shall shut up and read the paper, it’s usually the case that TW has thought of something I haven’t
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February 27th, 2010 at 5:36 pm
How are your thoughts precluded by Williamson’s? If he thinks (H1…) >= (H2…), this does not preclude the case that (H1…) > (H2…). Are you not simply making a stronger assertion?
The trouble with the mathematical interpretation of probability is that it is only what it is because we chose to define it that way, but from that perspective, that P(H2…) is a subset of P(H1…) of course only implies P(H1…) <= P(H2…) for an arbitrary probability measure.
This seems like a classic case of a limit. We all agree that P(H1…n) <= P(H2…n) true (where <= is a statement that N implies P(H2…n) - P(H1…n) < p. Would disagreeing with this logic not be equivalent to disagreeing with the statement that 0.999… = 1?
February 27th, 2010 at 5:40 pm
Sorry, I made some mistakes:
How are your thoughts precluded by Williamson’s? If he thinks (H1…) <= (H2…), this does not preclude the case that (H1…) < (H2…). Are you not simply making a stronger assertion?
The trouble with the mathematical interpretation of probability is that it is only what it is because we chose to define it that way, but from that perspective, that P(H2…) is a subset of P(H1…) of course only implies P(H1…) <= P(H2…) for an arbitrary probability measure.
This seems like a classic case of a limit. We all agree that P(H1…n) <= P(H2…n) true (where <= is a statement meaning N implies P(H2…n) - P(H1…n) < p. Would disagreeing with this logic not be equivalent to disagreeing with the statement that 0.999… = 1?
February 27th, 2010 at 5:50 pm
This thing keeps cutting off my post in random places…
This seems like a classic case of a limit. We all agree that P(H1…n) <= P(H2…n) true, whether in Williamson’s sense or in your stronger assertion. As we take n to infinity, the difference in probability becomes arbitrarily small (but positive).
February 27th, 2010 at 5:51 pm
However, no matter what positive real number we fix (call it p), there exists a natural number N such that n > N implies P(H2…n) - P(H1…n) < p. Would disagreeing with this logic not be equivalent to disagreeing with the statement that 0.999… = 1?
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