5.3 Exercise 13 — Applied Probability

A physician thinks the odds are about 2 to 1 that a patient has a certain disease. He seeks the “independent” advice of three specialists. Let H be the event the disease is present, and $A , B , C$ be the events the respective consultants agree this is the case. The physician decides to go with the majority. Since the advisers act in an operationally independent manner, it seems reasonable to suppose $\left\{A , B , C\right\} ci \left|\right. H$ and $ci \left|\right. H^{c}$ . Experience indicates

P \left(\right. A \left|\right. H \left.\right) = 0 . 8 , P \left(\right. B \left|\right. H \left.\right) = 0 . 7 , P \left(\right. C \left|\right. H \left.\right) = 0 . 75

P \left(\right. A^{c} \left|\right. H^{c} \left.\right) = 0 . 85 , P \left(\right. B^{c} \left|\right. H^{c} \left.\right) = 0 . 8 , P \left(\right. C^{c} \left|\right. H^{c} \left.\right) = 0 . 7

What is the probability of the right decision (i.e., he treats the disease if two or more think it is present, and does not if two or more think the disease is not present)?

Pfeiffer, Applied Probability, 5.3 Problems on Conditional Independence, exercise 13

AI Tutor