3% of the country has a disorder. However, the health institute recently developed a test for the disorder that has a 97% "true positive" rate (the probability that a person will test positive given that they have the disorder) and a 2% false positive rate (the probability that a person will test positive given that they do NOT have the disorder). If you take the test (independently) twice simultaneously and it shows that you got positive on one result and negative on another, what's the probability you actually have the disorder?
I've tried to solve this question for a couple of hours now, but I've had no luck. It seems like a Bayes' Theorem problem, but it seems like there are three "given" events to condition on. I'm not completely sure how to approach this problem. Can someone please help me? I'm not entirely sure how to utilize the fact that 3% of the country has a disorder.
I thought it might be the following:
$$\frac{0.03(0.97 \cdot 0.03 + 0.03 + 0.97)}{0.03 \cdot (0.97 \cdot 0.03 + 0.03 \cdot 0.97) + 0.97 \cdot (0.02 \cdot 0.98 + 0.98 \cdot 0.02)} \approx 0.043902,$$
but I'm really not sure.