If there is an infinite sequence of coinflips (for simplicity, assume that heads or tails are the only possible outcomes for each coinflip, and that this is a fair coin), is it possible for the coinflip to come up heads? I think so, but how do we prove it?
This issue bears relevance to the Eternal Society Argument against an infinite past. Roughly (in the paper the Eternal Society Paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).
Now imagine the Eternal Society has the following Annual Coin-Flipping Tradition: each year they flip a coin. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise they do not do the chant for that year.
The coinflips are probabilistically independent events, so any particular infinite permutation of coinflips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year for the Eternal Society practicing the aforementioned Annual Coin-Flipping Tradition. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.
Again, the coinflips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past for the Eternal Society engaging in the Annual Coin-Flipping Tradition. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.
Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:
- If an infinite past were possible, an Eternal Society would be possible.
- If an Eternal Society were possible, then scenario S1 would be possible.
- If S1 would be possible, then S2 would be possible.
- S2 is not possible.
- Therefore, an infinite past is not possible.
One could deny premise (1) particularly since that seems to be the most vulnerable premise, but as the Eternal Society Paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).
One objection I’ve seen is that it’s just not possible for a coin to come up heads infinitely many times, such that S1 is possible (since both heads and tails coming up infinitely many times is possible) but S2 is not (it’s impossible for an infinite sequence of heads to happen). The reasoning behind this may stem from a confusion of the law of large numbers or thinking that if the probability of heads is 50% for each trial, an infinite sequence of such trials leads to a probability of 0, and a probability of 0 means that it’s impossible (this is not true; any infinite particular permutation of coinflips where both heads or tails has a “probability of 0” as the number of trials goes to infinity; so this “probability of 0” reasoning would imply that no outcome is possible for an infinite sequence of fair coinflips). Regardless, can we mathematically prove that an infinite sequence of heads is possible under the conditions of the Eternal Society Paradox? We can.
To define some terms (with the caveat that different sources may define these terms slightly differently): in probability a random experiment like flipping a fair coin is called a trial. In the case of an individual coinflip for the scenario used here, the outcomes are binary meaning that there are only two possible outcomes (heads or tails). Mutually independent trials are where the probability of given trial’s outcome is unaffected by the outcomes of other trials, including any combination of the outcomes of the other trials. A sample space is the set of all possible outcomes of a random experiment.
In the case of an infinite sequence of coinflips, each trial has exactly two possible outcomes, each trial is mutually independent (in the sense that the probability of the outcome is unaffected by the outcomes of other trials), and the probability is the same in each trial. Some additional assumptions to make the reasoning a bit easier to follow if nothing else: each candidate outcome for a trial (heads or tails) has a nonzero probability, and any outcome with a nonzero probability is possible. Let H represent “heads” and T represent “tails.”
Recall that for each coinflip, both H and T have nonzero probabilities and thus both are possible. So the fact that the probability of given trial’s outcome is unaffected by the outcomes of other trials entails that which outcomes are possible for each trial is also unaffected by the outcomes of other trials.
So for each trial (and thus each sequence element in the infinite sequence of coinflips):
|(a)||both outcomes (H or T) are possible; and|
|(b)||which outcome is possible is unaffected by the outcomes of other trials.|
Given the aforementioned conditions, since every element in the sequence has the property of heads or tails both being possible values regardless of the values of the other sequence elements, this permits any infinite sequence of the binary values.
To illustrate, suppose we have a fair coin and the coinflips are mutually independent in the sense I described earlier, and there are an infinite number of coinflips. Can the first coinflip be heads? Yes, since for each sequence element, H or T is a possible outcome, which would include the first coinflip (confer (a)). Given that the first coinflip is heads, can the second coinflip also be heads? Yes, since the possible outcomes of the second coinflip is unaffected by the outcomes of other coinflips (confer (b)); hence both H and T can be used here. If the first two coinflips are heads, can the third one be also? Yes, because the possible outcomes of the third coinflip is unaffected by the outcomes of other coinflips (confer (b)), hence both H and T can be used here, and so on ad infinitum for all of the remaining sequence elements. And since the outcomes of heads is arbitrary here (e.g., we could just as well have used T, H, T for the first three sequence elements, and then choose whatever binary sequence we wish for the remaining sequence elements) this can be generalized so that the sample space consists of all infinite binary sequences.
But since the set of all possible outcomes consists of all infinite binary sequences, this would include a sequence of coinflips coming up H for each trial.