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Probability: Fallacies, Myths and Puzzles: Main Page

Why clever people cannot understand Bayes Theorem and what you can do about it (show an event tree)

The following is a classic problem that is easily solved by Bayes Theorem (it is the same problem we posed here).

"One in a thousand people has a prevalence for a particular heart disease. There is a test to detect this disease. The test is 100% accurate for people who have the disease and is 95% accurate for those who don't (this means that 5% of people who do not have the disease will be wrongly diagnosed as having it).

If a randomly selected person tests positive what is the probability that the person actually has the disease?"

The usual solution using the Bayes formula is given here (at the bottom of the page).

But there is a problem. Most people (including very clever ones like doctors and lawyers) cannot understand or follow the solution when presented in the normal 'formulaic' way. There are a number of reasons for this. Fear of mathematics is one and another is the inability of most people to understand abstract probabilities. For example, for the Bayes theorem solution you start with the prior probability of a person having the disease being equal  0.001. But most people have trouble understanding what a probability of 0.001 means and how, for example it differs from a probability of 0.0001. There are high court judges and top surgeons who would not understand that an event with probability of 0.001 is 10 times more likely than an event with probability 0.0001. But what people do understand is that if there is a one in a thousand chance of a person having the disease then in a set of 100,000 people about 100 will have the disease. So the simple trick to improve understanding is to use whole numbers in a contextualised set of people (rather than abstract probabilities) . Then you can present the equivalent of the Bayesian argument mechanically (using what is called an event/decision tree) like this:

event tree

What we have done here is consider the different alternatives (as in the formulaic approach), but in a way that makes it absolutely clear how we arrive at the correct posterior probability.
You can think of this as a formal version of this animation.  

Norman Fenton

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