A hospital identifies patients with an average false positive rate of 0.01. If it checks 31333 patients and returns a total of 357 positive results, then what's the expected number of true positives?
I thought it would just be 357 - 0.01 * 357 = 353.43, but this is wrong. I'm also confused because I don't see why the 31333 patients part is relevant.
Can someone please help me?
Let $C$ represent the event that a randomly selected patient has a condition that the test is intended to detect. Let $T$ represent the event that a randomly selected patient tests positive for the condition. Then a false positive is the event $T \mid \bar C$, i.e., a patient without the condition, when tested, yields a positive result.
You are given $$\Pr[T] = \frac{357}{31333},$$ and $$\Pr[T \mid \bar C] = 0.01.$$ You want to compute $$\Pr[T \mid C],$$ the probability of a true positive. To this end, we write $$\Pr[T] = \Pr[T \mid C]\Pr[C] + \Pr[T \mid \bar C]\Pr[\bar C].$$ Since $\Pr[C] + \Pr[\bar C] = 1$, we have $$\Pr[T \mid C] = \frac{\Pr[T] - \Pr[T \mid \bar C](1-\Pr[C])}{\Pr[C]} = 0.00139374 + \frac{1}{100 \Pr[C]}.$$ Without knowing the underlying prevalence of the condition, we see that it is not possible to uniquely determine the desired conditional probability, and therefore we also cannot determine the expected number of true positives.