How do you interpret conditional probability when two events are switched? Before I pose my question, I want to emphasize that I am not seeking a homework help or steps on how to derive the answer, for I already know the solution, and how to get it. What I am seeking is, how to  interpret  conditional probabilities when the dependent events are switched.


Suppose in a given population, $0.01$ people have a certain disease. A diagnostic test assesses whether a person has a disease. The probability that a healthy person is falsely diagnosed as having the disease is $0.05$, and the probability that someone unhealthy is diagnosed as healthy is $0.2$. What is the probability that a person has a disease given positive result?
  

I've already worked this out, and I got  $P(D|P) = 0.139$ , which is the correct answer. Now, here's what I am having a hard time interpreting. I feel that the value doesn't seem to reflect reality. Considering that  $P(P|D)$ is $0.80$ , common sense would tell me that there would be high chance of a person having a disease when a person is diagnosed positive. What are your thoughts?
 A: A good way to think about it intuitively is to consider a sample.
You have 2000 people. 20 of them are sick. If you tested the other 1980 of them, 99 of them would test positive. If you tested the 20 sick people, 16 of them would test positive. So the positive results in the sample consist of 16 sick people and 99 healthy people. Even though most healthy people test negative, there are so many more healthy people than than there are sick people that they dominate your sample. This occurs unless the false positive rate is very low (comparable to or less than the sickness rate). 
Indeed you can try redoing this calculation where you replace the false positive rate with $0.005$. You should see dramatic improvement.
A: A test might be very reliable, but if it tells you something very unlikely, you should factor in how unlikely that thing is when you're deciding whether or not to trust the test.

The Bayesian in the comic is factoring in the unlikeliness of the sun having exploded, while the frequentist is not.
