Understanding Base Rate Fallacy: A Lesson in Probability

Have you ever heard someone say, "I'm pretty sure I can tell who's guilty just by looking at them"? It may sound harmless, yet it points to a great common mistake in reasoning called base rate fallacy. It’s an important concept in statistics and probability that, once grasped, can help people make far better decisions in daily life.

What is Base Rate Fallacy?

Base rate fallacy occurs when we neglect to take the overall importance of an event in favor of specific details. In other words, our main mistakes are that we often pay too much attention to particular details and pay too little attention to broader statistical realities. Thus, we make faulty conclusions. This can be seen in the example posed at the start: the statement , "I'm pretty sure I can tell who's guilty just by looking at them" reflects the common human instinct to make judgements through intuition or superficial traits rather than solid, statistical information. But how does this relate to the study of mathematics? Let’s explore this with a more concrete example below.

Example: The Medical Test

Suppose there is a disease which affects every 1 of 1000 people. There exists a medical test that is accurate 90 percent of the time: it identifies with complete accuracy 90 percent of the people who have the disease (true positives). On the other hand, the test also identifies with perfect accuracy 90 percent of the people who don't have the disease (true negatives).

Now, let's assume that you took the test, and it turned out positive. That sounds pretty scary, right? But let’s take a closer look at the math behind this and put meaning into what that actually implies.

1. Base Rate: Only 1 out of every 1,000 people has the disease. So, the base rate of the disease is 0.1%, or 1 in 1,000.

2. True Positives: If you turn out positive, the probability of your actually having the disease is not just about how accurate the test is. We have to take into account those who don't have the disease.

3. False Positives: Since 1,000 are taking this test and 999 don't actually have the disease, it means that for healthy patients, too, this test will still wrongly identify 90% of them as positive. That is 899 testing positive incorrectly.

Now, we want to find the real probability that you have the disease given that you tested positive. Bayes' Theorem tells us:

Where we define:

P(Disease | Positive) as the probability of having the disease after a positive test;

P(Positive | Disease) as the probability of testing positive if you have the disease (0.9);

P(Disease) as the base rate of the disease, (0.001);

And P(Positive) as the overall probability of testing positive.

To find P(Positive), we have to take into account both true positives and false positives:

Plugging in the numbers:

P(Positive | No Disease)=0.1 (10% false positive rate)

P(No Disease)=0.999 

Doing the math:

P(Positive) = (0.9 × 0.001) + (0.1 × 0.999) = 0.0009 + 0.0999 = 0.1008

Now plug this back into Bayes' Theorem:

So, even after testing positive, there’s actually only about a 0.89% chance of actually having the disease. That is a classic base rate fallacy: if you focus on the positive test without considering the low base rate of the disease, you greatly overestimate the danger. To summarize, this table illustrates the example clearly when there are 10 thousand people:

	Test positive	Test negative
Actually positive	9	1
Actually negative	999	8991

As you can see, despite there being 1008 people testing positive for the disease, only 9 people have it. In this case, re-taking the test once or twice after testing positive would be required to more accurately determine whether one actually has the disease.

Why Does It Matter?

Understanding base rate fallacy is important not only in the medical sector but also other areas, like finance, legal arguments, and even in making decisions in everyday life. For example, consider hiring, where selection committees often anchor onto specific individual accomplishments rather than assessing the general quality of applicants that have been recruited.

The base rate fallacy serves as a good reminder to closely scrutinize the information we receive. The introduction of some basic concepts of probability into our thought processes allows for decisions based more on reason. So, the next time you hear a bold claim, take your time and look beyond the surface at the overall context. The numbers might just tell a different story!

Works Cited

Scribbr. (2023). Base Rate Fallacy. Available at: https://www.scribbr.com/fallacies/base-rate-fallacy/ (Accessed: April 1, 2024).

Bar-Hillel, M. (1980). The base-rate fallacy in probability judgments. Available at: https://www.sciencedirect.com/science/article/abs/pii/0001691880900463 (Accessed: March 30, 2024).

Hoffman, B. (2024). The base rate fallacy: what it is and how to overcome it. Available at: https://www.forbes.com/sites/brycehoffman/2024/05/31/the-base-rate-fallacy-what-it-is-and-how-to-overcome-it/ (Accessed: April 3, 2024).

The Decision Lab. (2023). Base Rate Fallacy. Available at: https://thedecisionlab.com/biases/base-rate-fallacy (Accessed: April 1, 2024).