philosophy

Some people may find all the other interpretations and definitions of probability are somewhat unintuitive: surely the probability of something happening is not really a "degree of belief", but something objective about the world that would be the same even if there were no human (or alien) beings around to believe it?

Probability theory was one of the last major areas of mathematics to be developed; its beginnings are usually dated to correspondence between the mathematicians Fermat and Pascal in the 1650's concerning some interesting problems that arose from gambling.

Cromwell's Rule refers to a principle that you should not give probabilities of 1 to any event that is not demonstrable by logic to be true, and never to give probability 0 to any event unless it can be logically shown to be false - see page 91 of Understanding Uncertainty by Dennis Lindley. It comes from Cromwell's appeal to the Church of Scotland to 'think it possible you may be mistaken'.

The frequency interpretation of probability is fairly self explanatory, in that it defines a probability as a limiting frequency. If we throw a (fair) die often enough, then we will eventually find that each number comes up about a sixth of the total time. The longer we continue to throw the die, the nearer the result will come to the ideal value of 1/6 for each number. The probability of an event is then defined by the relative frequency, as the throws continue to an idealized infinity.

We have seen in The logical view: probability as objective degree of belief that "logical" probability, defined as a degree of belief which it is rational and objective to hold given the available evidence, can give rise to some contradictory results. However, in the 1920s Frank Ramsey in Cambridge [1] and Bruno de Finetti in Italy [2] independently proposed a radically different interpretation of probability which avoids these paradoxes by asserting that probability is a subjective degree of belief.

References

The view of probability as an objective degree of belief was developed in the early 20th century by people such as Harold Jeffreys and the the young John Maynard Keynes [1], and was later adopted by the influential philosopher of science, Rudolf Carnap [2]. This view is not widely held these days, either by statisticians or philosophers, though there seems to be something of recent revival (see for example Williamson 2005 [3].

References

Due to our ignorance about the outcome of, say, a cast die, and because there is no indication for us to think one outcome more likely than any other, we must give them all an equal probability.

One of the curious features of probability is that in everyday speech it is used in two rather different circumstances: first, when there is some essential randomness, and second, when our uncertainty is due to lack of knowledge.

We use phrases like "the probability of this coin coming up heads is 1/2", and "the odds on Manchester United winning their match are 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.