Test results are positive. What are the odds you have the virus? Surprise, intuition is wrong!

Bayes’s Theorem

This is pretty much a direct application of Bayes’s Theorem:

P(A|B) = P(A) * P(B|A) / P(B)

P(A) is the probability of event A (probability of having the disease, which is 0.01)

probability table

P(A|B) = P(A) * P(B|A) / P(B) = 0.01 * 0.99 / 0.0198 = 0.5

If you test positive, the likelihood of actually having the virus is 50%! That’s very counter-intuitive. You would expect it’s around 99%. What happened?

chances vs infection rate

How about real-world testing?

Does that mean testing is useless when rates are low? Actually no.

  • the overall infection rate is very low (1%), and
  • you’re picking test subjects completely at random (picking completely random people out of the general population, like literally pulling first name / last name out of a hat)

How about negative test results?

I wrote a follow-up which deals with the negative results. Read it here:

Credits and code

The images are free to use, via Unsplash. Credit links:

sick.chance <- function(inf.rate, test.rel = 0.99, tot.pop = 10000) {
sick.pop <- tot.pop * inf.rate
healthy.pop <- tot.pop - sick.pop
pos.sick <- sick.pop * test.rel
neg.healthy <- healthy.pop * test.rel
pos.healthy <- healthy.pop - neg.healthy
pos.total <- pos.sick + pos.healthy
chance <- pos.sick / pos.total
infection.rate <- 1:100 / 100
chances <- lapply(infection.rate, sick.chance)
plot(infection.rate, chances, col = 'blue'); grid()



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Florin Andrei

Florin Andrei

Graduated Physics. Engineer in the computer industry. Working on my Master’s degree in Data Science.