Many of the things that statisticians and others investigate involve uncertainty. What will be the UK level of unemployment in a year’s time? If I take this drug my doctor has prescribed, will my health improve? If I drink a glass or two of red wine every day, will I live a longer or shorter time than if I don't drink wine at all? Will it rain tomorrow? The mathematical tool that is generally used to deal with such uncertainty is called probability.

Probability is a way of expressing the uncertainty of an event in terms of a number on a scale. The most common way, among statisticians at least, of expressing this uncertainty is on a scale going from 0 to 1, where impossible events are given a probability of 0 and events that will certainly happen are given a probability of 1.

Other events, that might or might not happen, are given probabilities at intermediate events on the scale. So an event that is as likely to happen as not is given a probability halfway along the scale, at ½ or 0.5. An event that is pretty likely to happen, but could possibly not happen, might have a probability of 0.95.

Other scales are used for probabilities. Sometimes they are expressed on a percentage scale, where impossible events have a probability of 0%, events that are certain get a probability of 100%, an event as likely as not to happen has a probability of 50% and so on. Bookmakers (and statisticians in some contexts) usually express uncertainty in terms of odds rather than probability.

Probability in action [Image squacco under CC-BY-NC licence]

If a horse-racing expert says that the odds on a particular horse winning a particular race are 1 to 2, he or she means that the chance of the horse not winning is twice as big as the chance of the horse winning. Expressing this on a probability scale going from 0 to 1, the probability that the horse will win the race is 1/3, and the chance that it doesn’t win is 2/3.

Probabilities obey various mathematical rules, many of which are quite simple and straightforward. For instance, tomorrow it will either rain or not rain. If the Met Office gives the probability of rain tomorrw as, let's say, 0.2 (that is, 20%), then then the probability that it won’t rain is 1 – 0.2 which comes to 0.8. In general, if the probability of an event is p, the probability that the event won’t happen is 1 – p. Using this and many other mathematical rules, a large body of mathematical theory about probabilities has been built up over several centuries.

Probability theory has been used to refine our understanding of random and chance occurences in the world, a subject in which people’s intuitions often lead them astray. If we want to use this body of theory to tell us something useful about the world, we need to have an understanding of what a probability means in terms of things in the real world. Actually, philosophers, statisticians and probability theorists have made this link between the maths and the world in several different ways. One common way, probably the most common, is as follows.

Suppose I’m going to toss a coin, and I say that the probability it will come up Heads is ½. OK, this means that a Head is as likely as a Tail, but what does that mean more precisely? Imagine that, instead of tossing the coin just once, I keep on tossing it again and again. In 100 tosses, I wouldn’t be surprised if the number of Heads wasn’t exactly 50. It might be 48 or 55, but it wouldn’t be too far from 50. In 1000 tosses, again, I wouldn’t expect exactly 500 Heads, but the proportion of heads would be very close to half. If the probability of a Head is really ½, then as I keep tossing again and again, the proportion of Heads would tend to get closer and closer to a half.

This long-run meaning of probability is all very well, but it doesn’t make much sense in contexts where things cannot be repeated. If a horseracing expert says that a particular horse’s probability of winning a particular race is ½, then it is hard to imagine the same horse running exactly the same race again and again and counting up how often it wins. The expert may well mean instead that he or she would be prepared to bet on the horse winning if the odds on offer were better than evens, but would not be prepared to bet if the odds were worse than evens. There’s no notion of repeating the race behind this sort of thinking, so it can be applied more widely than the long-run idea. But it has an inevitable subjective aspect. A different horseracing expert might give a different probability for the same horse winning the same race.

Probability forecasts for weather don’t explicitly relate to betting, but some of the ideas behind them are the same. Suppose the Met Office says that the probability of rain tomorrow in your region is 20%. They aren’t really talking about long-run repetitions of tomorrow. Tomorrow’s only going to happen once. They also aren’t saying that it will rain in 20% of the land area of your region, and not rain in the other 80%. No, forecasts like this are considerably less subjective than most horseracing tips, in that they are based on extensive observations and complicated computer models of the weather, but fundamentally they are just a way of expressing that, even with all that technology, tomorrow’s weather is uncertain. One can be more precise about the actual uncertainty by saying that the chance of rain is 20%, rather than just using words and saying “it might rain” or “there’s some chance of rain”.

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Thinking in terms of probability can often help to make more sense of situations involving chance, where our intuition can lead us astray. An example is the notorious Monty Hall problem.