Anyone speculating on the result of the European Union (EU) referendum can refer to either opinion polls or prediction markets. On the morning of June 20, polls put the probability that Remain will win at 52 per cent, while prediction markets estimate 75 per cent. That is a big difference.
When you toss a fair coin, the probability that it will come down heads is 50 per cent. That means that if you toss it many times, you will get either heads or tails half the time. But what do people mean when they say that the probability of Remain is 52 per cent or 75 per cent? The referendum is a one-off event and the outcome will be either Remain or Leave. A probability for Remain of 52 per cent cannot mean that if the referendum were held 100 times, the result would be Remain on 52 occasions.
Because the mathematics of probability is powerful and well understood, people who talk about uncertainty are keen to frame their discussion in probabilistic terms. But it is not obvious that the analogy works.
The pollsters who put the probability of Remain at 52 per cent make the following calculation: They assume that the people who have given answers to the pollsters are truthful, and a random sample (adjusted for differential response) of those who will actually vote tomorrow. They find that just over 50 per cent of the population will vote Remain.
Then they use statistical techniques to compute the errors involved in random sampling. That calculation tells them that if Remain is slightly ahead in their population, there is a 52 per cent chance that Remain is ahead in the population as a whole.
Those who use betting markets to compute probability approach the problem quite differently. They assume everyone has a "subjective probability" assessment of the result. You will bet on Remain at odds of evens if your subjective probability that Remain will win is 60 per cent, but bet on Leave if your subjective probability of Remain is only 40 per cent. The odds in the betting market reflect the amounts of money placed on each outcome. So they represent an average of everyone's subjective probability, weighted by the cash behind each assessment.
Which probability is right?
Both 52 per cent and 75 per cent are correct answers to different questions. The pollsters' answer is based on the voting intentions of respondents and the bookies' answer on their customers' assessment of the voting intentions of other people.
But this does not resolve the question of what either statement of probability means. When the Intergovernmental Panel on Climate Change states that it is 99 per cent probable that human influence has caused global warming, what they mean by their own definition is that they think it is extremely likely that human influence has caused global warming. The number conveys no additional information whatever.
When President Barack Obama held the crucial meeting to authorise the 2011 raid on Osama bin Laden's compound in Pakistan, his advisers' estimates of the probability that Osama was there ranged from 10 per cent to 95 per cent.
Mr Obama reportedly responded: "In this situation, what you started getting was probabilities that disguised uncertainty as opposed to providing you with more useful information." Summing up, he said: "This is 50-50. Look guys, this is a flip of the coin. I can't base the decision on the notion that we have any greater certainty than that."
The President did not mean that the probability that Osama was there was 0.5; still less that the most difficult decision of his presidency was based on the flip of a coin. What he meant was that, in the face of radical uncertainty, decision makers must act even when they simply do not know - without the aid of a pseudo-scientific numerical crutch.
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