If the multiverse and many-worlds don't exist, how should we interpret probabilities? 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?
Thanks
 A: 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.
A: 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.)
A: 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.
