The frequency of the Flu disease in the population is 1 in 10,000. A test that
checks if a person is infected with the disease is $99\%$ accurate on both sick and healthy
individuals. One takes the test and gets a positive response (test says he is infected). What
is the probability that he is actually infected?
My reasoning was as follows:
C: correct test
I: incorrect test
D: desease
HD: have desease
$P(C) = 0,99$
$P(I) = 0,01$
$P(D) = 0,0001$
$P(I) \to \ ?$
$P(HD) = P(C)P(D|C) + P(I)P(D|I)$
$P(HD) = P(C)P(D) + P(I)P(D)$ Are independents
$P(HD) = P(D)[P(C)+P(I)]$
$P(HD) = P(D)$
Is correct my reasoning? Why if not.