Recall that physicians are now using Bayesian reasoning (including prior, conditional, joint and posterior probabilities)
in the medical diagnostic phase of patient examinations. Whenever a patient has an undiagnosed disease, the
examining physician initially develops prior probabilities of diseases the patient might have. Typically, these
probabilities are assigned using the "subjective method," and are usually a function of such things as the physician's
view of various relevant environmental factors such as the occurrence of epidemics, geography, recent weather
patterns, and climate.

Suppose a physician believes a patient has one of two possible diseases, denoted D 1 and D2 with P(D1) = 0.30
and P(D2) = 0.70.

Suppose also that medical research has established the probability associated with each of three symptoms
(denoted S1, S2, and S3) that may accompany the two diseases. That is, suppose that, given diseases D1 and D2,
the probabilities that the patient will have symptoms S1, S2, or S3 are as follows: P(S1|D1) = .25; P(S2|D1) =
.15; P(S3|D1) = .65; P(S1|D2) = .10; P(S2|D2) = .15; P(S3|D2) = .20.

After a certain symptom is found to be present, the medical diagnosis may be aided by finding the revised
probabilities of each particular disease. Compute the posterior probabilities of each disease given the following
medical findings.
a. The patient has symptom S1. D1 _________ and D2 _________
b. The patient has symptom S2. D1 _________ and D2 _________
c. The patient has symptom S3. D1 _________ and D2 _________
d. For a patient with symptom S1 in part (a), we also find symptom S3. The posterior probabilities are: D1 _________
and D2 _________
First, we make a contingency table as shown below:
D1
D2
Total
S1
0.25 * 0.3 = 0.075
0.1 * 0.7 = 0.07
0.145
S2
0.15 * 0.3 = 0.045
0.15 * 0.7 = 0.105
0.15
S3
0.65 * 0.3 = 0.195
0.2 * 0.7 = 0.14
0.335
0.3
0.7
(a) (i) P(D1|S1) = P(D1 and S1)/P(S1) = 0.075/0.145 = 0.5172
(ii) P(D2|S1) = P(D2 and S1)/P(S1) = 0.07/0.145 = 0.4828
(b) (i) P(D1|S2) = P(D1 and S2)/P(S2) = 0.045/0.15 = 0.30
(ii) P(D2|S2) = P(D2 and S2)/P(S2) = 0.105/0.15 = 0.70
(c) (i) P(D1|S3) = P(D1 and S3)/P(S3) = 0.195/0.335 = 0.5821
(ii) P(D2|S3) = P(D2 and S3)/P(S3) = 0.14/0.335 = 0.4179
(d) Really not sure of this. Tried a lot, but unable to fix it. I am really very sorry.