Bayes theorem and conditional probability can be used in medical diagnosis. Prior probabilities of diseases are based on the physicians assessment of factors such as

Bayes’ theorem and conditional probability can be used in medical diagnosis. Prior probabilities of diseases are based on the physician’s assessment of factors such as geographic location, seasonal influence, and occurrence of epidemics. Assume that a patient is believed to have one of two diseases, denoted D1 and D2, with P(D1) = 0.60 and P(D2) = 0.40 and that medical research shows a probability associated with each symptom that may accompany the diseases. Suppose that, given diseases D1 and D2, the probabilities a patient will have symptoms S1, S2, or S3 are as follows:

After finding that a certain symptom is present, the medical diagnosis may be aided by finding the revised probabilities the patient has each particular disease. Compute the posterior probabilities of each disease for the following medical findings.
a. The patient has symptom S1.
b. The patient has symptom S2.
c. The patient has symptom S3.
d. For the patient with symptom S1 in part (a), suppose that symptom S2 also is present. What are the revised probabilities of D1 and D2?

Symptoms 0.15 0.80 0.10 0.15 0.15 0.03 Disease D, PS ID