Given a population, the probability event that a person is carrying a virus is $A$. When someone takes a test, the probability event that the test will come back positive is $B$. If asked to calculate the probability that a test will give a falsely-positive result, would you evaluate $P(A^c \mid B)$ or $P(B \mid A^c)$ ?
When presented the problem, I calculated the second probability, but looking at the answers, I found out that the prof. calculated the first one, and most of the time those two will be different. I can't imagine it's up to everyone to interpret it as they understand it, so, is it the question a bit unclear, or am I missing something?
$\Pr(A^c\mid B)$ is for when it is given a test showed that the person tested positive, what the probability that the person does not carry the virus, i.e. we are only asking the question to people who tested positive.
$\Pr(B\mid A^c)$ is for when it is given the person does not carry the virus, what the probability they test positive, i.e. we are only asking the question to people who do not carry the virus.
$\Pr(A^c\cap B)$ is the probability that a person both tests positive and does not have the virus, i.e. we are asking all people regardless of status.
The way you phrased your question, "If asked to calculate the probability that a test will give a falsely-positive result" sounds to me as asking what the probability of having both a positive result and not carrying the virus and asking this question to the population at large, hence $\Pr(A^c\cap B)$.
If you wanted to ask "calculate the probability that a test will give a falsely-positive result given that ____" that is an entirely different question and will give different answers depending on what is put in the blank.