# Neglecting the Base Rate

**what is the probability of a random person being infected, given a positive test result?** Assuming test sensitivity of 95% and 5% of the population are infected.

The correct answer is 20%. If your answer was much higher, then you have committed a **base rate fallacy**, and here is why. Under uncertainty, we tend to make a quick judgment based on a presupposition. In psychology, this is called the representativeness heuristic.

Base rate fallacy is reasoning without taking prior knowledge into account. The solution is the **Bayes theorem**. The beauty of this approach relies on conditional probability, the essence of probabilistic reasoning.

For example: we have a test for infection with a sensitivity of 0.95, which means the test returns a 95% positive result on the infected population. Suppose further the test specificity is 0.80, which means the test return 80% negative results on the healthy population.

With a 0.05 prevalence (base rate), **what is the probability of a random person being infected, given a positive test result?**

Using Bayes Theorem:

P(H|E) = P(H) P(E|H) / P(E)

P(Infected | Positive) = P(Positive | Infected) P(Infected) / P(Positive | Infected) + P(Positive | Healthy)

`0.95 * 0.05 / (0.95 * 0.05 + 0.20 * 0.95 ) = 0.20 = 20%`

So, what is the probability of a random person being healthy, given a positive test result?

P(Healthy | Positive) = 1 — P(Infected | Positive)

`1–0.20 = 0.80 = 80%`

The takeaway, **there is an 80% probability of a random person being healthy, given a positive test result.** Committing a base rate fallacy as a decision-maker will result in disasters! I’ll leave this to your imagination. Now, you know. It would be easier for you to identify this bias when it happens in real life.