Your question highlights an important issue with respect to different interpretations of probability. The frequentist has great difficulty attaching a probability to a single event. Strictly speaking, there cannot be a frequency of a single event, so frequentists tend to wave their arms a lot and claim that we can consider something to be an instance of a long run of trials in principle. Such a claim is not plausible in general. What is the probability that Hillary Clinton will become the next US president? (This is written in 2016 with the primaries still in progress.) Such an event, if it happens, will be unique: there cannot be a long run frequency of it. If you try to assess its probability as a frequency you cannot get any sensible value for it. What is the frequency of a female president? Zero. What is the frequency of a democrat president? About a half? What is the frequency of a democrat president given that the previous president was a democrat? Less than a half. Whatever frame of reference you choose will give you a different answer. Frequentists might try to tough it out and say that such an event does not have a probability, because it lacks a frequency, but this is implausible. You can bet on Hillary to win (or not win) and betting odds imply probabilities.
The virtue of the epistemic approach to understanding probability is that the derivation of the probability calculus can be made without reference to frequencies or possibility spaces, but by starting from simple assumptions about how decisions are made under uncertain information, and about what constitutes a bad decision. We all have to make decisions, and we almost always have to do so with imperfect information. Frequently we make bad decisions, and often this happens because we have failed to quantify the uncertainty properly. One approach to deriving probability is to take a bet as a paradigm case of a decision under uncertainty, and a Dutch book as a paradigm case of a bad decision. (A Dutch book is a combination of bets that results in you losing, no matter what happens.) A fairly remarkable result, first proved by Bruno de Finetti, is that from this consideration only, if you wish to avoid making bad decisions, your calculus of uncertainty must conform to the probability calculus. Of course, this is a fairly limited concept of 'bad decision' - it corresponds to nothing more than maximising expectation value, and as many theorists have pointed out, it is not always the best strategy to maximise expectation value. Nevertheless it is a powerful concept: it shows that we can derive probability theory from decision theory.
This provides you with the answer to your question: probability is used to make decisions because properly understood it is exactly that quantity that allows us to make decisions under uncertainty without falling into straightforward errors of judgement.