### What is Probability?

Probability is a quantification of uncertainty. We use probability words in our everyday discourse: impossible, very unlikely, 50:50, likely, 95% certain, almost certain, certain. This suggests a shared understanding of what probability is, and yet it has proved very hard to operationalise probability in a way that is widely accepted.

Uncertainty is subjective

Uncertainty is a property of the mind, and varies between people, according to their learning and experiences, way of thinking, disposition, and mood. Were we being scrupulous we would always say "my probability" or "your probability" but never "the probability". When we use "the", it is sometimes justified by convention, in situations of symmetry: tossing a coin, rolling a dice, drawing cards from a pack, balls from a lottery machine. This convention is wrong, but useful -- were we to inspect a coin, a dice, a pack of cards, or a lottery machine, we would discover asymmetry.

Agreement about symmetry is an example of a wider phenomenon, namely consensus. If well-informed people agree on a probability, then we might say "the probability". Probabilities in public discourse are often of this form, for example the IPCC's "extremely likely" (at least 95% certain) that human activities are the main cause of global warming since the 1950s. Stated probabilities can never be defended as 'objective', because they are not. They are defensible when they represent a consensus of well-informed people. People wanting to disparage this type of stated probability will attack the notion of consensus amongst well-informed people, often by setting absurdly high standards for what we mean by 'consensus', closer to 'unanimity'.

Abstraction in mathematics

Probability is a very good example of the development of abstraction in mathematics. Early writers on probability in the 17th century based their calculations strongly on their intuition. By the 19th century mathematicians were discovering that intuition was not good guide to the further development of their subject. Into the 20th century mathematics was increasingly defined by mathematicians as 'the manipulation of symbols according to rules', which is the modern definition. What was surprising and gratifying is that mathematical abstraction continued (and continues) to be useful in reasoning about the world. This is known as "the unreasonable effectiveness of mathematics".

The abstract theory of probability was finally defined by the great 20th century mathematician Andrey Kolmogorov, in 1933: the recency of this date showing how difficult this was. Kolmogorov's definition paid no heed at all to what 'probability' meant; only the rules for how probabilities behaved were important. Stripped to their essentials, these rules are:

1. If A is a proposition, then Pr(A) >= 0.
2. If A is certainly true, then Pr(A) = 1.
3. If A and B are mutually exclusive (i.e. they cannot both be true), then Pr(A or B) = Pr(A) + Pr(B).

The formal definition is based on advanced mathematical concepts that you might learn in the final year of a maths degree at a top university.

'Probability theory' is the study of functions 'Pr' which have the three properties listed above. Probability theorists are under no obligations to provide a meaning for 'Pr'. This obligation falls in particular to applied statisticians (also physicists, computer scientists, and philosophers), who would like to use probability to make useful statements about the world.

Probability and betting

There are several interpretations of probability. Out of these, one interpretation has emerged to be both subjective and generic: probability is your fair price for a bet. If A is a proposition, then Pr(A) is the amount you would pay, in £, for a bet which pays £0 if A turns out to be false, and £1 if A turns out to be true. Under this interpretation rules 1 and 2 are implied by the reasonable preference for not losing money. Rule 3 is also implied by the same preference, although the proof is arcane, compared to simple betting. The overall theorem is called the Dutch Book Theorem: if probabilities are your fair prices for bets, then your bookmaker cannot make you a sure loser if and only if your probabilities obey the three rules.

This interpretation is at once liberating and threatening. It is liberating because it avoids the difficulties of other interpretations, and emphasises what we know to be true, that uncertainty is a property of the mind, and varies from person to person. It is threatening because it does not seem very scientific -- betting being rather trivial -- and because it does not conform to the way that scientists often use probabilities, although it does conform quite closely to the vernacular use of probabilities. Many scientists will deny that their probability is their fair price for a bet, although they will be hard-pressed to explain what it is, if not. Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.

### Converting probabilities between time-intervals

This is the first in an irregular sequence of snippets about some of the slightly more technical aspects of uncertainty and risk assessment.  If you have a slightly more technical question, then please email me and I will try to answer it with a snippet. Suppose that an event has a probability of 0.015 (or 1.5%) of happening at least once in the next five years. Then the probability of the event happening at least once in the next year is 0.015 / 5 = 0.003 (or 0.3%), and the probability of it happening at least once in the next 20 years is 0.015 * 4 = 0.06 (or 6%). Here is the rule for scaling probabilities to different time intervals: if both probabilities (the original one and the new one) are no larger than 0.1 (or 10%), then simply multiply the original probability by the ratio of the new time-interval to the original time-interval, to find the new probability. This rule is an approximation which breaks down if either of the probabilities is greater than 0.1. For example

### 1-in-200 year events

You often read or hear references to the ‘1-in-200 year event’, or ‘200-year event’, or ‘event with a return period of 200 years’. Other popular horizons are 1-in-30 years and 1-in-10,000 years. This term applies to hazards which can occur over a range of magnitudes, like volcanic eruptions, earthquakes, tsunamis, space weather, and various hydro-meteorological hazards like floods, storms, hot or cold spells, and droughts. ‘1-in-200 years’ refers to a particular magnitude. In floods this might be represented as a contour on a map, showing an area that is inundated. If this contour is labelled as ‘1-in-200 years’ this means that the current rate of floods at least as large as this is 1/200 /yr, or 0.005 /yr. So if your house is inside the contour, there is currently a 0.005 (0.5%) chance of being flooded in the next year, and a 0.025 (2.5%) chance of being flooded in the next five years. The general definition is this: ‘1-in-200 year magnitude is x’ = ‘the current rate for eve

### Detectable impacts of Climate Change in the UK; a new review for the next Climate Change Risk Assessment

2022 was another year of “unprecedented” weather. Provisional figures indicate that it was the warmest so far recorded, with almost every month hotter than average. Much of the country had a notably mild New Year, despite the cold snap in mid-December. This was preceded by the third warmest autumn on record , and that by a scorching summer, with the hottest day ever recorded in July. But summer’s heat waves were also accompanied by a rise in the number of daily deaths across the country. People around the world are becoming increasingly more aware of events like these, and their impact in the UK is particularly concerning amidst the ongoing cost-of-living, energy, and NHS crises. Aerial view of the Wennington wildfire, London, 19 July 2022. Source: Harrison Healey, Wikimedia Commons  (CC BY 3.0). Ahead of the Fourth UK Climate Change Risk Assessment (CCRA4), the Climate Change Committee (CCC) are asking what we know about the impact of past and present climate change on natural and