Question

Suppose that in order for the defendant to be convicted in a military court the majority of the nine appointed judges must enter a guilty vote. Assume that a judge enters a guilty vote with probability 0.2 or 0.8 if the defendant is innocent or guilty, respectively, independently of other judges. Assume also that 50% of the defendants in such trials are innocent.

(a) What proportion of all defendants is convicted?
(b) What is the proportion of correct verdicts?

Answer 1

a) 0.5

b) 0.8

Explanation:

Given:

P(Convicted | Innocent) = 0.2

P(Convicted | guilty) = 0.8

P(Innocent) = 50% = 0.5

P(Guilty) = 0.5

a) P(defendant is convicted)

= P(guilty and convicted) + P(innocent and convicted)

= P(Guilty) × P(Convicted | guilty) + P(Innocent) × P(Convicted | Innocent)

= 0.5 × 0.8 + 0.5 × 0.2

= 0. 4 + 0.1

= 0.5

b) P(correct verdict)

= P(guilty and convicted) + P(innocent and not convicted)

= P(Guilty) × P(Convicted | guilty) + P(Innocent) × (1 - P(Convicted | Innocent) )

= 0.5 × 0.8 + 0.5 × ( 1 - 0.2)

= 0.4 + 0.4

= 0.8