I'm going to do some writing about discrete probability theory. Probability is an extremely important area of math. We encounter aspects of it every day. It's also a very poorly understood area - it's one that we see abused or just fouled up every day.
I'm going to focus on discrete probability theory. What that means is that we're going to look at things where the space containing the things that we're going to look at contains a countable number of elements. The probability of getting a certain sequence of coin flips, or of getting a certain hand of cards are described by discrete probability theory. On the other hand, the odds of a radioactive isotope decaying at a particular time requires continuous probability theory.
Before getting into the details, there's one important thing to mention. When you're talking about probability, there are two fundamental schools of interpretetation. There are frequentist interpretations, and there are Bayesian interpretations.
In a frequentist interpretation, when you say the probability of an event is 0.6, what you mean is that if you were to perform a series of experiments precisely reproducing the event, then on average, if you did 100 experiments, the event would occur 60 times. In the frequentist interpretation, the probability is an intrinsic property of the event. For a frequentist, it makes sense to say that there is a "real" probability associated with an event.
In a Bayesian interpretation, when you say that the probability of an event is 0.6, what you mean is that based on your current state of knowledge about the event, you have a 60% certainty that the event will occur. In a strict Bayesian interpretation, the event doesn't have any kind of intrinsic probability associated with it. The specific event that you're interested in either will occur, or it won't. There's no real probability involved. What probability measures is how certain you are about whether or not it will occur.
For example, think about flipping a fair coin.
A frequentist would say that you can flip a coin many times, and half of the time, it will land on heads. So the probability of a coin flip landing on the head of the coin is 0.5. A Bayesian would say that the coin will land either on heads or on tails. Since you don't know which, and you have no other information to use to be able to make a better prediction, you can have a certainty of 0.5 that it will land on the head of the coin.
In the real world, I think that most people are really somewhere in between.
I think that all but the most fervent Bayesians do rely on an intuitive notion of the "intrinsic" probability of an event. They may describe it in different terms, but when it comes down to it, they're using the basic frequentist notion. And I don't think that you can find a sane frequentist anywhere who won't use Bayes theorem to update their priors in the face of new information - which is the most fundamental notion in the Bayesian interpretation.
One note before I finish this, and get started on the real meaty posts. In the past, when I've talked about probability, people have started stupid flamewars in the comments. People get downright religious about interpretations of probability. There are religious Bayesians, who think that all frequentists are stupid idiots who should be banished from the field of math; likewise, there are religious frequentists who think that Bayesians are all a crop of arrogant know-it-alls who should be sent to Siberia. I am not going to tolerate any of that nonsense. If you feel that you cannot read posts on probability without going into a diatribe about those stupid frequentists/Bayesians and their deliberately stupid ideas, please go away and don't even read these posts. If you do go into such a diatribe, I will delete your comments without any hesitation.