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If the multiverse of eternal inflation and the many-worlds of quantum mechanics don't exist, how should we interpret the meaning of probabilities? If there is only one copy of the universe out there, what does it mean to say a radioactive nucleus has a 1/2 probability of decaying? Does it mean if we have 1,000,000 similar radioactive nuclei, 500,000 $\pm$ 1,000 of them will decay? How do we make sense of the $\pm$ 1,000 error bar?

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Even if many-worlds don't exist we can still imagine them. You can assume that you have $100^{100}$ $ in your account and start calculating taxes, can't you? – Pratik Deoghare May 30 '11 at 9:09
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The notion of probability was invented well before discovering multiverse nature of our multiverse nature. – Vladimir Kalitvianski May 30 '11 at 9:22
@Vladimir Kalitvianski: When the multiverse nature of the universe was discovered? – Fabian May 30 '11 at 15:50
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@Fabian When they started to speak about it without shame. – Vladimir Kalitvianski May 30 '11 at 16:09

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This is essentially a philosophical rather than a physical question. (I don't mean that as a pejorative statement, by the way, just a descriptive one.) There are different philosophical approaches regarding the meaning of probability. Broadly speaking, some people think of probability in a frequentist sense, meaning that probabilities refer to the frequency of outcomes in an ensemble. Other people think of probability as a description of our knowledge, rather than a description of the system itself. In this way of thinking about things, often called "Bayesian," probabilities don't require one to postulate or even to imagine ensembles.

Personally, I think that the frequentist attitude is self-evidently absurd. When I look at the weather report and see that there's a 70% chance of rain tomorrow, I am not required to, and certainly do not, imagine an ensemble of infinitely many Earths, 70% of which will contain rain tomorrow. Rather, this statement simply means that I don't know whether it'll rain or not, but I'll be somewhat more surprised if it doesn't than if it does. E.T. Jaynes's book is a standard (and in my opinion excellent) manifesto in favor of this attitude towards probability.

Weather forecasts are not quantum probabilities (at least not entirely, or in any obvious way), but I don't think that matters. I see no reason I can't regard all probabilities in exactly the same way.

(By the way, the terms frequentist and Bayesian also refer to different sets of statistical techniques. Although that usage is related to the philosophy-of-probability usage, they're not the same. In particular, I am not claiming (nor do I believe) that frequentist statistical techniques are absurd or invalid. I actually use both sets of techniques in my own work, depending on which is more convenient for any given problem.)

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Without especially advocating frequentism, but defending its non-absurdity, weather services can and do track their forecasting accuracy. Of all the hours in which a forecasting algorithm predicts that there is a 70% chance of rain, say, I suppose that after the event there should be 70% of those hours in which rain was recorded. Anything significantly different suggests a need to change the algorithm. Is that absurd? Bayesianism's invocation of a prior always seems magical to my prejudices. A priori and a posteriori can both be given houseroom, just as your last paragraph admits. – Peter Morgan May 30 '11 at 17:23
I'm certainly not arguing that probabilities shouldn't, on average, correspond to frequencies, in situations where repeated trials occur. Of course they should! What I claim is absurd is the idea that probabilities, fundamentally, mean frequencies. An even sharper example of the issue: surely it makes sense to talk about the probability that, say, Barack Obama will be reelected in 2012. But in that case there is obviously no interpretation in terms of an actually-existing ensemble, and (to me) the idea that we need to postulate an imaginary ensemble to understand this probability is silly. – Ted Bunn May 30 '11 at 17:29
@Ted, this won't fit in comments! If one takes a unique event like Barack Obama's election, Bayesian updating is no more useful than frequentism. The post facto updated probability cannot be applied to any future question. If it is claimed that the updated probability can be applied to some future question, then that begins to inductively define what the ensemble is. Choosing to consider every event to be unique makes the methods of Science inapplicable (can't repeat any experiment!), so we construct ensembles that are identifiably similar enough in whatever way we find useful. – Peter Morgan May 30 '11 at 18:03
Not that I'm a polling expert, but to make a justifiable claim that the probability that Obama will win the 2012 election is X%, one has to measure something. Polls do. How much has been spent on the early campaign by the opponents, how friendly, commanding, effective, whatever, do polls find Obama and the parties to be, etc., etc., and how do those numbers correlate with election results in the past? There is evidence if one looks widely enough, but of course different commentators will use different statistical methods that effectively take various DoFs to be differently significant. – Peter Morgan May 30 '11 at 18:18
I'm not sure I understand your point. I'm not claiming that the question of interpretation leads to any practical difference at all between how one does or should use probabilities in practice. I'm just talking about what probabilities fundamentally mean. The original question was whether one needs something like many-worlds in order to interpret probabilities. The answer would be yes, if the actual existence of an ensemble were necessary for talking about probabilities. But it's not. – Ted Bunn May 30 '11 at 18:32
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Dear student, yes, probability always has the same interpretation. If you observe 1 million events - they don't have to be similar (each of the events may be totally different!) - and each of them has the probability 1/2 to produce a "Yes" answer, then you will get 500,000 plus minus 1,000 "Yes" outcomes.

There is an error margin because the predictions are probabilistic and they make complete sense. This is nothing specific to quantum mechanics. If you throw dice 1 million times, you get a number greater than three 500,000 times plus minus 1,000 as well. This makes complete sense and is calculable. What is hard to make sense of is your question.

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You answer isn't helpful to student's (although arguably not physics-related) question. – David M. R. Nov 16 '11 at 10:32
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The multiverse interpretation is an ensemble interpretation. Frequentism is actually something different with only one universe, but many trials of essentially the same kind of experiment in that one single universe, and in that one single universe, the relative ratios approach the probability as the number of trials goes to infinity. Bayesianism is the subjective belief interpretation.

Everyone else left out the most appealing interpretation, the propensity interpretation favored by the philosopher of science Karl Popper. This is the closest to the common sense meaning, but leads to circular definitions. Circular definitions are not a flaw, but a sign that something is foundational. Hence the mystery. How to tie in propensity theory to quantum mechanics is an interesting open question.

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Circular definitions are not a flaw, but a sign that something is foundational. Hence the mystery. That sounds awfully suspect. – dbaseman Nov 7 '12 at 5:19

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