keith_devlin's picture
Mathematician; Executive Director, H-STAR Institute, Stanford; Author, Finding Fibonacci
Base rate

The recent controversy about the potential dangers to health of the back-scatter radiation devices being introduced at the nation's airports and the intrusive pat-downs offered as the only alternative by the TSA might well have been avoided had citizens been aware of, and understood, the probabilistic notion of base rate.

Whenever a statistician wants to predict the likelihood of some event based on the available evidence, there are two main sources of information that have to be taken into account:

1. The evidence itself, for which a reliability figure has to be calculated;

2. The likelihood of the event calculated purely in terms of relative incidence.

The second figure here is the base rate. Since it is just a number, obtained by the seemingly dull process of counting, it frequently gets overlooked when there is new information, particularly if that new information is obtained by "clever experts" using expensive equipment. In cases where the event is dramatic and scary, like a terrorist attack on an airplane, failure to take account of the base rate can result in wasting massive amounts of effort on money trying to prevent something that is very unlikely.

For example, suppose that you undergo a medical test for a relatively rare cancer. The cancer has an incidence of 1% among the general population. (That is the base rate.) Extensive trials have shown that the reliability of the test is 79%. More precisely, although the test does not fail to detect the cancer when it is present, it gives a positive result in 21% of the cases where no cancer is present — what is known as a "false positive." When you are tested, the test produces a positive diagnosis. The question is: What is the probability that you have the cancer?

If you are like most people, you will assume that if the test has a reliability rate of nearly 80%, and you test positive, then the likelihood that you do indeed have the cancer is about 80% (i.e., the probability is approximately 0.8). Are you right?

The answer is no. You have focused on the test and its reliability, and overlooked the base rate, Given the scenario just described, the likelihood that you have the cancer is a mere 4.6% (i.e., the probability is 0.046). That's right, there is a less than 5% chance that you have the cancer. Still a worrying possibility, of course. But hardly the scary 80% you thought at first.

In the case of the back-scatter radiation devices at the airports, the base rate for dying in a terrorist attack is lower than many other things we do every day without hesitation. In fact, according to some reports, it is about the same as the likelihood of getting cancer as a result of going through the device.