What is Probability?

We use phrases like "the probability of this coin coming up heads is 1/2", and "the odds on Manchester United winning their match is 2 to 1", and "the chance of dying of cancer is 30%". But what do these numbers actually mean? There are fundamentally different views about this, which can lead to very different ideas about how to deal with uncertainty.

The mathematics of probability may be sorted out, but the underlying concepts are not straightforward. Maybe that's why the study of probability only started 350 years ago, which is relatively recent compared to other mathematical ideas.

The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements about when probabilities can be used. If I throw a die, look at the result but don't show you, what is the probability (for you) that it is a 6? In one interpretation it is still 1/6, since you don't know the result, but in another interpretation the probability is either 1 or 0, since the die shows either a 6, or it doesn't.

The other sections are roughly chronological:
The development of probability introduces the origins of probability theory and asks why it has only been developed so recently.

Gambling ca. 1800 (from Wikimedia Commons)

The classical approach shows the way probability has been interpreted early on, as a measure of our ignorance, in particular by the 18th - 19th century mathematician Laplace.
The logical approach interprets probability as a rational and objective degree of belief, and sees probabilistic reasoning as an extension of logical reasoning, with clear and indisputable answers.
The subjective approach in contrast interprets probability as a subjective degree of belief: there is no such thing as an objective probability; instead a probability is determined by what we are willing to bet on something happening. It can be shown that, if we do not want to be vulnerable to losing money, our betting odds follow the mathematical laws of probability.
The frequency approach considers probability to be relative frequency with which events occur in the long run: if we toss a fair coin long enough, then the relative frequency of heads against tails will tend to be 1/2.
The propensity approach considers probabilities to be inherent in the experimental set up, and was developed (at least in one of its incarnations) mainly to iron out a shortcoming with the frequency approach, namely that it does not allow us to talk of the probability of events that occur only once.