Homework: Bayesian Thinking

 I took a quick test to determine how good I am at Bayesian thinking.

 (1) A doctor assigns you at random for a test for disease D. 1 in 10,000 people in the USA have disease D. The test is 99% accurate, meaning that the probability of a false positive is 1%. The probability of a false negative is 0.  You test positive. What is the new probability that you have disease D?

I first thought that since the test has a high accuracy and that there are no false negatives, that the probability that I have the disease after testing positive would be pretty high. However, I was surprised to see that the probability of having the disease given that I tested positive is just 1%, which is very low.  Just because there are no false negatives and has a very low false positive rate does not mean it is accurate because we do not know for sure if someone has a disease. Since the disease is very rare, it is more likely for anyone to not have the disease. This skews the calculations for a positive test to be a false positive, even if the false positive rate is very low. 

(2) You left the USA to go to another country, M, recently, then returned to the USA. You took the test for disease D after you returned and tested positive. 1 in 200 people in the USA who visited M recently return with disease D. What is your probability of having the disease?

My intuition from before I worked it out on paper was that the probability would increase with the given conditions. After doing the work, my intuition was correct because the probability of having the disease became 3%.

This test was revealing because it showed that my intuition was very wrong for the first question. I saw that the test had low rates of false positives and negatives and I equated that with the test being accurate. However, since we don’t actually know if someone has the disease or not, it is hard for us to understand what false positive or negative really means.