Chance and Explanation

Flip a coin a thousand times and it will come up heads about half the time. Repeat the process and the results will be similar. Continue the repetitions and you might get some divergence from half now and then, but they will be rare. You won’t get a run of 1000 heads or tails if you spend your whole life flipping coins.

This is a notable regularity. Notable regularities invite (or even require) explanations. Can we explain the stability of the frequency or results?

One might say: the frequency is can be explained by objective chance. The coin has a 50/50 objective chance of landing heads, and objective chances produce relatively stable frequencies.

One problem with this answer is that objective chances don’t require stable frequencies. It is (conceptually) consistent with the coin having an equal chance of landing heads and tails that it land heads all the time. So how does the coin’s .5 objective chance of landing heads explain the stability of frequency?

One might say: the objective chance doesn’t require stable frequencies, but it makes them more probable. This might be understood in two ways. First, we might read it as saying that there is a high objective chance that stable frequencies result. This reading just pushes the problem back a level. 

Why think that something with a high objective chance of occurring explains why it occurs?

Alternatively, we might read it as saying that we should believe that chances result in stable frequencies, even though they needn’t. However, this odd fact itself requires explanation. Why is it that we should believe that objective chances make certain beliefs about frequencies rational? The Principal Principle, which says that we should align our personal subjective probability assignments with known chances, is certainly true, but it is most plausibly regarded as a constraint on what a chance would have to be. It seems like the way that chances explain frequencies should come before, and partly explain, why they make certain beliefs about frequencies rational.

Chance seems to be to be a dormitive virtue-esque property. We posit chances to explain frequencies, but chances give us no explanatory grasp on frequencies. They are merely that-thing-whatever-it-is-that-explains frequencies. They are placeholders for a proper explanation.

How could we explain frequencies? We might explain frequency in terms of laws of nature that mandate that certain frequencies obtain.  The fact that we see certain patterns among heads and tails is that there is some law of nature that requires that those patterns occur. Get heads enough times in a row, and the universe says: “tails now”. It then becomes physically impossible for other patterns to predominate.

Instead, I think it is more promising to think about explain chances in terms of population sizes. Shuffle a pack of cards and draw six. They won’t be all face cards. Why not? Because there are many many more ways of drawing six cards with at least some non-face cards than there are drawing six face cards. There is nothing requiring that we draw non-face cards, but it is unsurprising that we don’t because the vast majority of possible draws don’t.

There are a couple of ways to implement this idea to explain objective chance. One way is to say that in the space of possible ways that the universe could progress, the majority of possible histories involve stable frequencies. This makes it is unsurprising that we find ourselves within a history with stable frequencies.

Something like this idea is used in statistical mechanics, and can perhaps explain macroscopic chancy events like coin-tosses. It is possible that something similar is going on at a lower level. So when we talk about chances of radium atoms decaying, perhaps we should explain this by saying in terms of the number of possible ways the atom could decay versus the number of ways it could fail to decay.

Another way of using this idea to explain frequencies is to say that in the space of actual observers, by necessity, the majority of observers see stable frequencies. Suppose we exist in a branching universe, and whenever a chancy event occurs, the universe splits off into a number of different branches. The number of branches where one result occurs, divided by the total number of branches, is the ‘chance’ of that result occurring. Then the vast majority of observers will observe stable frequencies, and we should not be surprised to find ourselves within the majority. Some versions of the Everettian interpretation might be able to explain quantum chanciness. If so, that seems to be a mark in their favor.

I don’t mean to say that these are the only ways that we might explain stability. But I am pulled toward thinking that there must be some explanation that is as good as the above proposals of why chancy events result in stable frequencies.