I'm wondering about the meaning of ascribing probabilities to the outcomes of nonrepeating events. As a concrete example, here in the UK we're pollsters are currently predicting the result of the referendum on UK's membership in the European Union. They say things such as "Britain will stay in the EU with 70% probability", but I just don't understand the intuition behind this statement. If we had a referendum every week, and if we could assume that the individual outcomes were independent, then this would mean that, in about 70% of the cases, people would vote to stay. In reality, however, we will actually have just one referendum, so there is no way to talk about 70% of the outcomes. Thus, the number "70%" seems completely arbitrary to me and it doesn't seem to carry much more information than, say, 42!
I assume that my question is related to What does actually probability mean? and the relationship between the frequentist and Bayesian interpretation of probability. I've read the Wikipedia page mentioned in that article, and I think I understand the frequentist interpretation and find it intuitive; however, it seems to me that this interpretation relies on events repeating. Moreover, I just can't get my head around the Bayesian interpretation. So I would really appreciate it if someone could clearly explain the following points:
Does it at all make sense to talk of probability of outcomes of one-off events? Maybe people just use the word "probability" without really paying attention to its formal meaning?
Am I right in thinking that the frequentist interpretation requires repeating events?
Could someone please explain or point me to a nice paper/book explaining the intuition behind the Bayesian probability?
Please note that I'm not asking which interpretation is right; I'd like to understand the merits of either interpretation independently. Many thanks in advance!