Conditional Probability for a Medical Test

conditional probabilitystatistics

There are two test for a disease: one is rapid and the other is slow. Given an infected individual, the rapid test will register positive 40% of the time, while the slow test will register positive 80% of the time: additionally, both tests will be positive 35% of the time.

  1. Given an infected person and that the rapid test measures a positive result, what is the probability that the slow test is also positive?

I assumed the tests were independent, giving just 80%. This is incorrect however.

I tried also P(slow) * P(rapid) = 0.80 * 0.40 = 0.32, which was also incorrect.

I also tried P(A|B) = P(A and B)/ P(B) leaving just P(A) = 0.8

I'm not sure how to incorporate the latter half of the information either. Is 35% referring to positive for all tests (infected or uninfected individuals), i.e. false-positive rate?

  1. Given an infected person and that the rapid test measures a negative result, what is the chance that the slow test is positive.

Same problem here. Why would the results of a rapid test affect the results of a slow test–shouldn't it just be 80%?

Best Answer

My reading is that for an infected person

  • there is a probability of $0.35$ both tests show positive
  • there is a probability of $0.4-0.35=0.05$ the fast test shows positive and the slow test negative
  • there is a probability of $0.8-0.35=0.45$ the fast test shows negative and the slow test positive
  • there is a probability of $1-0.35-0.05-0.45=0.15$ both tests show negative

So the answers are:

  1. $\frac{0.35}{0.35+0.05}=0.875$ for the probability that the slow test is positive, given an infected person and that the rapid test measures a positive result
  2. $\frac{0.45}{0.45+0.15}=0.75$ for the probability that the slow test is positive, given an infected person and that the rapid test measures a negative result
Related Question