So you think you know the odds

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The human mind finds probability difficult, – no matter what FT readers think. Probabilistic models are often applied in situations of fundamental uncertainty, where the problem – far more complex even than a quiz game – is not sufficiently well defined to be amenable to such analysis. But to disregard probabilistic reasoning altogether is also a certain source of error.

The thesis I presented in last week’s column – that the human mind finds probability difficult – was amply confirmed by the response that filled my mailbox and the editor’s.

Contestants on the Monty Hall show choose from three closed boxes, one of which contains the keys to a car. After you have made your choice, Monty opens one of the two remaining boxes, which is empty. Should you alter your choice? The answer is yes. He might have spoiled the game by opening a box that contained the car keys, but he didn’t. Monty might have been blind drunk and could as easily have opened your box by mistake – but he didn’t. It does not matter what Monty knows, or is thinking. All you need know is that the box Monty opens is empty.

Why is it so difficult to confirm the answer to what ought to be a simple logical problem? Even in this simple case, it is hard to define the issue with sufficient precision to reach an analytic solution. What does Monty know? What do the producers of the show expect? Does the mathematical representation describe the question accurately?

That is why correspondents who wanted to substitute common sense for probability theory were sometimes closer to the mark. In a group selected at random to be tested for a rare disease suffered by one person in 1,000, even if the test is 99 per cent accurate it is still not very probable that someone who tests positive has the disease. The chance that you have the disease is less than one in ten.

The reason the mathematical solution contradicts our intuition is because the problem posed is peculiar. People are not often selected completely at random for medical tests. Suppose you have been referred for testing by an incompetent doctor. Most of the people he refers do not have the disease and most patients who are indeed suffering from it are not referred. But so long as the doctor’s failure rate is less than 99 per cent you should worry about a positive result. Even a small element of non-randomness in selection transforms the problem. It is now more likely than not that you have the disease.

Perhaps the reason we are not very good at dealing with probabilities is that there are not many real life situations in which knowledge of probability is useful. Our lack of skill in managing uncertainty misleads us in two opposite ways. One is to use probability theory where it is inappropriate. Probabilistic models are often applied in situations of fundamental uncertainty, where the problem is not sufficiently well defined to be amenable to such analysis. This is the fundamental flaw of the “value at risk” modelling widely used in the financial sector. These techniques would reduce risks to low and predictable levels if the models were correct and complete descriptions of the world. But no model whose parameters are based on historic data represents, or could represent, the full range of uncertainties in the economic and political environment. Institutions that use these techniques will suffer large losses, not because of “six sigma events” – extreme outcomes anticipated with very low probabilities – but because events occur that were not anticipated at all.

But to disregard probabilistic reasoning altogether is also a certain source of error. When you play roulette, you must expect to lose: the predictions of probability theory are confirmed by the empirical observation that casino operators are mostly prosperous.

The strongest argument for learning about probability theory is that people who don’t understand it lose money to people who do. The demonstration of this by the Cambridge philosopher Frank Ramsey and the Chicago economist Jimmie Savage is the reason why probabilistic reasoning became the dominant approach to uncertainty in the 20th century.

On those African savannahs where modern brains evolved, rational calculating individuals won money at dice from hunters whose irrational exuberance brought in the wild game that fed the tribe. The same phenomenon today explains, not just the race track and Las Vegas, but why derivatives markets are so profitable – for the people who run them.

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