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## Cambridge University Press Probability is a familiar idea - but what does it really mean?

# What exactly is probability, anyway?

It is important - but surprisingly difficult - to be clear about what we mean by probability. In fact, probability can mean very different things to different people!

Some popular options for the definition of the probability of an event include:

• Symmetry: ‘The number of outcomes favouring the event, divided by the total number of outcomes, assuming the outcomes are all equally likely.’ This is the definition usually taught in schools as theoretical probability, but it is rather circular as it depends on ‘equally likely’ being defined. And, it can only be used in nicely balanced situations such as dice, cards or lottery tickets, or when, to use a classic example, picking a coloured sock at random from a drawer. It does not apply, for example, to the probability that you will have a heart attack or stroke in the next ten years.
• Frequency: ‘The proportion of times, in the long run of identical circumstances, that the event occurs.’ This is the idea of an observed relative frequency tending to a true probability after sufficient repetitions. This can be fine for situations where there are lots of repeats, but does not seem applicable to unique situations, such as your risk of a heart attack.
• Subjective: ‘My personal confidence that an event will occur, expressed as a number between 0 and 1. When the event either occurs or not, my assessment will be rewarded or penalised according to an appropriate ‘scoring rule’.’ This definition is one way of formalising the idea that probabilities are purely personal judgements based on available evidence and assumptions.

You may also wish to read this article from the website ‘Understanding Uncertainty’, which describes some of the different perspectives in more detail.

What does probability mean to you?