A Classical Example of The Base Rate Fallacy
As part of the Information Cascades topic, we learned about the Base Rate Fallacy. In fact, the first time when I heard about this phenomenon was in the Introduction to Psychology class that I took a year ago. In this class, I learned about a famous psychology experiment designed by Bar-Hillel, in which participants are given the following base rate information. There are two cab companies in a city: a “Green” company that has green cabs and a “Blue” company with blue cabs. It is given that 85% of the cabs in the city are green and 15% are blue. Then, the experimenter describes a scenario in which a cab was involved in a hit-and-run accident at night. There was one witness to the incident, and they identified the cab as blue. In order to assess the reliability of the witness’s testimony, the court tested their ability to differentiate between blue and green cabs at nighttime. It turned out that the witness accurately distinguished the two colors 80% of the time but confused them 20% of the time. The participants in the experiment are then asked to estimate the probability that the cab involved in the accident was blue, given the witness’s testimony.
Most people responded that this probability is 80%. However, everyone who gave this answer is enacting the base rate fallacy. The right answer is, in fact, 41%. We can compute this using Bayes’ rule. Let B be the event of the cab being blue, and W be the event that the witness identifies the cab as blue. Then the probability that the cab is blue given that the witness identifies it as blue is as follows:
The figure below is a good illustration of the problem. Suppose that these are the cabs that the witness was tested on. There are 15 blue cabs and 85 green ones, which aligns with the base rate information that we are given. The bordered region indicates the cabs that the witness identifies as blue, and the rest are identified as green. We see that 80% of the cabs were identified correctly, and 20% were identified incorrectly. However, only 12 out of the 29 cabs that the witness identified as blue are actually blue.
This experiment demonstrates that many people can be subject to the base rate fallacy. It is tempting to assign greater value to a piece of individuating information and ignore the base rate information, and this has a psychological explanation. This phenomenon can occur in many different situations, and unfortunately, it can propagate misinformation and extreme beliefs. Hence, the base rate fallacy phenomenon is useful to keep in mind when we hear people throwing numbers around to make a point.
Source: Bar-Hillel, M. (1980). The base-rate fallacy in probability judgments. Acta Psychologica, 44(3), 211–233. doi: 10.1016/0001-6918(80)90046-3