A particular test correctly identifies those with a certain serious disease 94% of the time and correctly diagnoses those without the disease 98% of the time. a friend has just informed you that he has received a positive result and asks for your advice about how to interpret these probabilities. He knows nothing about probability, but he feels that because the test is quite accurate, the probability that he does have the disease is quite high. You research and discover that 4% of men have the disease. What is the probability that your friend actually has the disease?

This item can be solved by the probability theorem called Bayes' theorem which states that the probability of event A occurring given event B is equal to,

P (A/B) = P (A) P (B/A) / P (B)

where P (B/A) is the probability that the test will yield positive if the person has the disease. P (A) is the probability will be present in any particular person which is equal to 0.04.

P (B) is the probability of positive result irrespective of whether the disease is present of not is calculated below.

P (B) = (0.94) x (0.04) + (0.06) (0.96) = 0.0952

Now, solving for P (A/B)

P (A/B) = (0.94) (0.04) / 0.0952 = 0.039

Thus, the answer is approximately 4%.