The Base Rate Fallacy occurs when we disregard or undervalue prior information when making a judgment on how likely something is. In particular, if presented with related base rate information (i.e. generic, general information) and specific information (information pertaining only to a certain case), the fallacy arises from a tendency to focus on the latter at the expense of the former.

For example, we are informed that someone is an avid book-lover, we might think it more likely that they are a librarian than a nurse. There are, however, many more nurses than librarians. In this example, we have not taken sufficient account of the base rate for the number of nurses relative to librarians.

Now consider testing for a medical condition, which affects 2% of the population. Assume there’s a test for this condition which will correctly identify them with this condition 95% of the time. If someone does not have the condition, the test will correctly identify them as being clear of this condition 80% of the time.

Now consider a test a random group of people. Of the 2% of patients who are suffering from the condition, 95% will be correctly diagnosed with the condition, whereas  of the 98% of patients who do not have the condition, 20% will be incorrectly diagnosed as having the condition (almost 20% of the population).

What this means is that of the 21.5% of the population (0.95 x 2% + 0.2 x 98%) who are diagnosed with the condition, slightly less than 2% (0.95 x 2% = 1.9%) actually are suffering from the condition, i.e. 8.8%.

Exercise

Consider testing for a medical condition, which affects 4% of the population. Assume there’s a test for this condition which will correctly identify them with this condition 90% of the time. If someone does not have the condition, the test will correctly identify them as being clear of this condition 90% of the time.

If someone tests positive for the condition, what is the probability that they have the condition?