Conditional probabilities can be useful in diagnosing disease. Suppose that three different, closely related diseases (A1, A2, and A3) occur in 25%, 15%, and 12% of the population. In addition, suppose that any one of three mutually exclusive symptom states (B1, B2, and B3) may be associated with each of these diseases. Experience shows that the likelihood of P(Bi /Ai) having a given symptom state when the disease is present is as shown in the following table. Find the probability of disease A2 given symptoms B1, B2, B3, and B4, respectively.
| Symptom State Bj |
Disease State Ai |
| A1 |
A2 |
A3 |
| B1 |
0.08 |
0.17 |
0.1 |
| B2 |
0.18 |
0.12 |
0.14 |
| B3 |
0.06 |
0.07 |
0.08 |
| B4(no symptoms) |
0.68 |
0.64 |
0.68 |