Question

Suppose also that the probability that a judge votes a guilty person innocent is 0.2, whereas the probability that the judge votes an innocent person guilty is 0.1.There are three judges who vote independently and correctly with probability p (either if the person is guilty or innocent). The decision is based on the majority of the judges. How large must p be so that correct decisions ( if 65 percent of the defendants are guilty) are made more frequently?

Answer 1

We are given that the probability that a judge votes a guilty person guilty = 0.8 and that the judge votes a innocent person innocent = 0.9 , also we are given that 0.65 of the defendants are guilty and 0.35 are innocent .

Now P is the probability that majority of the judges give correct decision i.e Guilty is voted guilty and innocent is voted innocent .

Therefore there will be total 8 cases when the correct decision is taken which are as follows :

1. 0.65*0.8*0.8*0.2 *3 ( 2 judges vote guilty and 1 not guilty ) ( *3 because there are 3 possibilites/cases )

2. 0.65*0.8*0.8*0.8  ( ALL judges vote guilty )

3. 0.35*0.9*0.9*0.1*3 ( 2 judges vote innocent and 1 guilty ) ( *3 because there are 3 possibilites/cases )

4. 0.35*0.9*0.9*0.9 ( ALL the judges voted innocent ) .

Thus P= 1*2*3*4= 0.9226