Difference between true positive and true negative in this case

Question

A patient takes a lab test and the result comes back positive. It is known that the test returns a correct positive result in only 98% of the cases and a correct negative result in only 97% of the cases. Furthermore, only 0.008 of the entire population has this disease.

Am i correct in understanding that a test correctly diagnoses an illness that is there 98% of the time. Hence misses the illness 2% of the time?

However for the correct negative, it is saying that it diagnoses that the illness is not there correctly 97% of the time. Hence states that the illness is there 3% of the time when it is not?

Answer 1

The answer to your questions is yes. To be more formal, let $D$ be the event the person has the disease, and let $P$ be the event the test result is positive, meaning that the test says the person has the disease.

Then $D^c$, the complement of $D$, is the event the patient is healthy. And $P^c$ is the event the test says the patient is healthy.

We are given certain conditional probabilities. The first one says that $$\Pr(P|D)=0.98.$$ The second says that $$\Pr(P^c|D^c)=0.97.$$ For the kind of problem you are likely to be asked, the second condition is best rewritten as $$\Pr(P|D^c)=0.03.$$ This says that on healthy people, the test gives a (false) positive with probability $0.03$.

Remark: We used one form of complement notation. There is a wide variety of notations for the complement of an event $A$, including $A'$.

Complement notation is potentially confusing. For example, one could easily end up writing $\Pr(P^c|D^c)=0.03$ instead of the correct $\Pr(P|D^c)=0.03$. Terms have a distressing tendency to look all alike.

It may be better in solving the problem to use $D$, say, for "has the disease," $H$ for "healthy," $P$ for "test result positive," and $N$ for "test result negative." It is important to choose notation that has a clear meaning to you. It is also important, in your answer, to define the meaning of the symbols you use. If you don't, it may upset/confuse the grader, and at least as importantly, it may lead to confusion on your part.