# The nature of probability

Kevin McConway, Emeritus Professor of Applied Statistics at The Open University, helps to explain the nature of probability and how weather forecasting and horse racing are unlikely partners when it comes to beating the odds.

As one of the top two performing weather forecasting centres in the world, Met Office forecasts are highly valued. Continuing improvements in accuracy with, for example, four day forecasts today being as accurate as a one day forecast back in the 1980s, enable the public and society to take a wider range of weather related decisions with more confidence. The chaotic nature of weather does mean that there are unavoidable limitations to what we can predict. However, by calculating the confidence in a weather forecast we aim to give people a clear picture of any uncertainties.

## Beating the odds

Weather forecasting and horseracing have more in common than you might think. Both involve predicting uncertain events. Will it rain on my wedding tomorrow? Will this horse win the next race? And there can be consequences of getting the prediction wrong - soaked guests, or lost money on bets. Nobody expects a racing tipster to make perfect predictions of all the winners - there's too much uncertainty. Weather, with its chaotic nature and many variables, is undoubtedly even more complex, and that adds to the potential uncertainty. Many people are familiar with expressing the uncertainty in the outcome of a horse race in terms of odds, and we can do something very similar with weather forecasts using probability, which expresses the chance of particular weather occurring.

Probability is a way of expressing the uncertainty of an event in terms of a number on a scale. One very common way of doing this is on a scale going from 0% to 100%, where impossible events are given a probability of 0% and events that will certainly happen are given a probability of 100%.

Other events, that might or might not happen, are given intermediate values on the scale. So an event that is as likely to happen as not is given a probability halfway along the scale, at 50%. An event that is pretty likely to happen, but could possibly not happen, might have a probability of 95%.

Other scales are used for probabilities. In horseracing, bookmakers usually express uncertainty in terms of odds rather than probability. If a horse-racing expert says that the odds against a particular horse winning a particular race are 3 to 1, he or she means that the chance of the horse not winning is three times as big as the chance of the horse winning. Expressing this on a probability scale from 0% to 100%, the probability that the horse won't win the race is 75% (3 in 4), and the probability that it will win is 25% (1 in 4).

If you're not familiar with the details, these different ways of expressing chances might look confusing, but there are calculation rules for going from odds like 3 to 1 to probabilities on a scale from 0% to 100%. It's just a matter of how the numbers are presented.

If we want to use probabilities to tell us something useful, we need an understanding of what a probability means in terms of the real world. Perhaps the most common way is as follows.

Suppose I'm going to toss a coin, and I say that the probability it will come up Heads is 50%. 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 1000 tosses, I'd expect something close to 500 Heads - not necessarily exactly 500, because coin tosses are always uncertain. But the 50% probability means I should get Heads on about half the tosses in the long run. - and if I don't, I can tell that the probability of a Head wasn't 50% after all.

This long-run meaning of probability is all very well, but it doesn't make so much sense in contexts where things cannot be repeated exactly. In horseracing, you can't imagine the same horse running exactly the same race again and again and counting up how often it wins. And when the Met Office gives a probability of rain for your region tomorrow, they aren't really talking about long-run exact repetitions of tomorrow. Tomorrow's only going to happen once.

There are other ways of linking probabilities and the real world, that don't involve the idea of long-run repetitions. But there's still a way of checking the probabilities against what really happened. In horseracing, if an expert says that a particular horse has an 80% chance (odds of 1 to 4) of winning a particular race, and the horse doesn't win, that doesn't mean the expert was wrong, because they didn't say the horse was sure to win. But if they give an 80% probability for lots of horses in lots of races, and actually only 30% of those horses win, something is wrong.

In weather forecasting, suppose the Met Office says that the probability of rain tomorrow in your region is 80%. They aren't saying that it will rain in 80% of the land area of your region, and not rain in the other 20%. Nor are they saying it will rain for 80% of the time. What they are saying is there is an 80% chance of rain occurring at any one place in the region, such as in your garden.

Forecasts like this are considerably more objective than most horseracing tips. They are based on extensive observations, complicated computer models, world-leading scientific techniques, super-computer technology and sometimes the input of highly qualified meteorologists. So a forecast of 80% chance of rain in your region should broadly mean that, on about 80% of days when the weather conditions are like tomorrow's, you will experience rain where you are. But there is still some expert judgement involved, and different forecasting organisations with different experts and different technology might well give different probabilities.

If it doesn't rain in your garden tomorrow, then the 80% forecast wasn't wrong, because it didn't say rain was certain. But if you look at a long run of days, on which the Met Office said the probability of rain was 80%, you'd expect it to have rained on about 80% of them. The Met Office checks their probability forecasts against what really happened in this way, and they do pretty well.

Fundamentally, though, these forecasts are just a way of expressing that, even with all that technology, tomorrow's weather is not completely certain. Some days we can be more certain than others, and one can be more precise about the actual uncertainty by saying that the chance of rain is 80%, rather than just saying "there's a good chance of rain". That can help you to decide whether to change your plans, depending how badly the wrong weather will affect you. But it still can't tell you for sure whether your parade will be rained off.

*Kevin McConway
Emeritus Professor of Applied Statistics, The Open University*

Last updated: Oct 9, 2016 9:10 AM