# probability

As of the 23rd May 2022 this website is archived and will receive no further updates.

understandinguncertainty.org was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

Many of the animations were produced using Flash and will no longer work.

## Screening for breast cancer

Since 1988, women over 50 in the UK have routinely been offered screening for breast cancer, even if they have no other symptoms, and in 2004/05 1.7 million women were screened. Those with a positive mammogram are recalled for further investigations, at considerable cost in anxiety, resources and pain and discomfort. But how many of these women really have breast cancer?

## The propensity approach

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?

## What are the chances?

In Why coincidences happen we saw how the chance of a rare event occurring to someone, somewhere, depended on the number of opportunities for it to occur.

## The development of probability

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.

## Risk

Risk has been studied from a variety of perspectives, which can most easily be categorised as the psychological approaches to risk and the social scientific approaches to risk [link - to come]. Both traditions are concerned with finding out more about how people percieve and understand risk, and how they react towards it, though they differ in their interpretations of what risk actually is, and in their ideas of what if anything should be done about their insights to risk.

## Cromwell's Law

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

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.

## Subjective degree of belief

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 Ramsey1926 and Bruno de Finetti in Italy DeFinetti1931 independently proposed a radically different interpretation of probability which avoids these paradoxes by asserting that probability is a subjective degree of belief.

## The logical view: probability as objective degree of belief

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 Keynes1921, and was later adopted by the influential philosopher of science, Rudolf Carnap Carnap1950. 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 Williamson2005.

## The classical approach

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.