Guessing the result of infinitely many coin tossesSeptember 22, 2008
What is the probability that an infinite number of coin tosses all land heads up? In a relatively recent analysis paper Tim Williamson argues, convincingly I think, that the probability must be 0. What I’m going to say here is to do with a related puzzle, and may shed some light on this question, although I haven’t fully absorbed the consequences. But either way, it strikes me as very weird, so any comments would be welcome.
The puzzle I’m concerned with involves guessing the results of infinitely many coin tosses. The setup is as follows:
For each n > 0, at hours past 12pm the following is going to happen: aware of the time, you are going to guess either heads or tails, and then I am going to flip a coin and show you the result so you can see if you are right or wrong. This process may have to be done at different speeds to fit it all in to the hour between 12pm and 1pm.
[Note how this differs from the Williamson sequence of coin tosses in that it is a backwards supertask.] The question I’m interested in is: how well can you do at this game? For example, can you, with absolute certainty, guess every result correctly? Intuitively, the answer is ‘no’, and unsurprisingly the answer to this question is indeed ‘no’. Can you adopt a strategy for guessing such that you are guaranteed to get at least one guess right? Intuitively the answer is still ‘no’ – no matter how unlikely, it is still possible that you always guess the wrong answer, and it is hard to see how adopting a certain way of guessing will get you out of this if you are extremely unlucky. Despite the intuition, however, the answer to this second question is in fact ‘yes’! There is a way of guessing such that you are guaranteed to guess right at least once. If that doesn’t strike you as weird yet, think about it a bit more before reading on.
Things get even weirder. It turns out that there’s a way of guessing such that, following it, you are completely guaranteed, no matter how the coins land, to guess all but finitely many of the tosses correctly. That is, if you follow this strategy, you are guaranteed to only make finitely many mistakes. Among other things, this means that at some point after 12pm you won’t have made any mistakes at all! You would have guessed an infinite sequence of coin tosses correctly. So it follows that there is a way of guessing such that it is guaranteed that if you follow it, you will guess the result of a fair toin coss correctly infinitely many times in succession.
The construction of the strategy is actually relatively simple (and it resembles the solution to this infinite hat problem closely.) Firstly we divide the space of all possible complete sequences of heads and tails the coin could land in into equivalence classes as follows. By a possible total sequence of heads and tails I mean a list describing the result of each coin toss at each time 1/n past 12. The sequences are divided into equivalence classes according to whether they agree at all but finitely many places in the sequence. I.e. let a ~ b iff a and b disagree about how the coin lands at most at finitely many places. Now, with the help of the axiom of choice, we can pick a representative sequence from each equivalence class – so we have a choice function that associates each equivalence class with an element it contains.
The stategy you should adopt runs as follows. At 1/n hrs past 12 you should be able to work out exactly which equivalence class the completed sequence of heads and tails that will eventually unfold is in. You have been told the result of all the previous tosses, and you know there are only finitely many tosses left to go, so you know the eventual completed sequence can only differ from what you know about it at finitely many places. Given you know which equivalence class you’re in, you just guess as if the representative of that equivalence class was correct about the current guess. So at 1/n hrs past 12 you just look at how the representative sequence says the coin will land and guess accordingly.
Now, since the representative sequence and the actual sequence of heads and tails that occurs are in the same equivalence class, they must only differ at finitely many places. So, if you guessed according to the representative sequence, you have only made finitely many mistakes.
Now this raises some further puzzles. For example, suppose that it’s half past twelve, and you know the representative sequence predicts that the next toss will be heads. What should your credence that it will land heads be? On the one hand it should be 1/2 since you know it’s a fair coin. But on the other hand, you know that the chance that this is one of the few times in the hour that you guess incorrectly is very small. In fact, in this scenario it is “infinitely” smaller in comparison, so that your credence in heads should be 1. So it seems this kind of reasoning violates the Principal Principle.