Bayes theorema question, two tests (one positive, one negative)

I have this question, and I would need help with the second part of it.

2 % of the population has had covid. The test has 97% true positive rate and 4% false positive rate.

Person takes two tests, first one is positive, and the second one is negative.

What is the probability of having the disease? 1) After the first test and 2) what it is after the second one, assuming the tests are independent and there are no systematic errors in testing?

My try so far:

Using Bayes theorema and placing given values into it. where: P(B|A): probability of getting positive test if you have the virus: 0,97 P(A): Probability of having the virus: 0,02 P(B): Probability of getting positive result: 0,0586 =(0,97 * 0,02) / 0,0586 = 0,3310 ≈ 33% prob.
I assume that I got right the first one.

But then I have no clue how to get the second. Because if it was too positive I think I would just use the same method than in 1. one but replace the 0,02 with 0,33. But how do I do it if its negative. I just don't get it, could you please help me to the right direction.


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Kav10 Kav10
  • Hi man, thank you! I will tip you coffee as a extra, if you can explain why you use in the second test 0,96 as the false negative and 0,03 the true negative, as they should be the opposite way by my logic, because originally the true negative is 0,96 and false negative rate is 0,03. That is the only thing that is bothering me still 😅 thanks!

    • Kav10 Kav10

      Hi man! Sure. See if this helps. In your question, the test has 97% true positive rate and 4% false positive rate. So, 1-0.97=0.03 is true negative and 1-0.04=0.96 is false negative. So, if the person has the disease, and test is negavite (in the second test), so the test needs to be false negative (0.96). Let me know if it still bothers you! :-)

  • okay, so It would then be totally different situation and calculation, if we knew that for sure the true negative rate would be 0,96? (person gets negative result if doesn't have the virus). Because I thought that it is the opposite percentage of false positive (0,04), and that was the part where I was wrong apparently😅

    • Kav10 Kav10

      Yeah, it would change how we put the numbers (e.g. percentages) in the formulas. It gets complicated with all that positive, negative, true, false, etc. and combinations.

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