Here is a probability question i have no clue about. "The Doctor's Dilemma. It is known that, among all....

....patients displaying a certain set of symptoms, the probability that they have a particular rare disease is 0.001. A test for the disease has been developed. The test shows a positive result on 98% of the patients who have the disease and on 3% of patients who do not have the disease.

The test is given to a particular patient displaying the symptoms, and it records a positive result. Find the probability that the patient has the disease. Comment on your answer."

It's a very different, not to mention tricky, question to me. how do i do it? please be detailed. i have only started with probability and right after simple questions on ''different colour discs'' the writer smacks me with this one. thanks.

Suppose such a test were given to the entire population. The probability of a positive result indicating the disease is (number of positive results with the disease)/(total positive results).

If the population is N, 0.001*N people have the disease and P1 = (0.001*N)*0.98 of them will test positive.

0.999*N people don't have the disease and P2 = (0.999*N)*0.03 of them will test positive.

So, the probability that a person testing positive actually has the disease is:

p = P1/(P1 + P2) = (0.001 * N * 0.98) / (0.001 * N * 0.98 + 0.999 * N * 0.03)

The factors of N cancel and you get:
p = 0.00098 / (0.00098 + 0.02997) = 0.031664

About 3.2%. The original problem is so rare that, in spite of the accuracy of the test, the 3% false positive rate in the healthy population far exceeds the diseased population.

I used a statistical program. It gives me

Sensitivity (prob. of true positive): 0.980
Prob. of false positive: 0.030

Specificity (prob. of true negative): 0.970
Prob. of false negative: 0.020

Prevalence: 0.500 (the standard setting on my program)