Consider a population of 100,000 people of which 5% have a disease, but without symptoms. There is a test available, which can screen for the disease. Denote having the disease by D, not having the disease by ND; testing positive by TP and testing negative by TN. Denote the probability of A conditional on B, P(A/B). The test is 90% accurate. This means (i) it falsely tests positive for those without the disease 10% of the time,
P(TP/ND)=0.1, and (ii) it also falsely tests negative for those with the disease 10% of the
time, P(TN/D)=0.1.
What is the probability of having the disease P(D/TP)?

Question

Consider a population of 100,000 people of which 5% have a disease, but without symptoms. There is a test available, which can screen for the disease. Denote having the disease by D, not having the disease by ND; testing positive by TP and testing negative by TN. Denote the probability of A conditional on B, P(A/B). The test is 90% accurate. This means (i) it falsely tests positive for those without the disease 10% of the time,
P(TP/ND)=0.1, and (ii) it also falsely tests negative for those with the disease 10% of the
time, P(TN/D)=0.1.
What is the probability of having the disease P(D/TP)?