Court Verdicts: Guilty, Not Guilty, Not Proven

[cloud360]

Homework Statement

A court returns three verdicts. ‘guilty’, ‘not guilty’ and
‘not proven’. 65% of the verdicts were ‘guilty’, 25% of the verdicts
were ‘not guilty’ and 10% were ‘not proven’.

When the court’s verdict is ‘guilty’ 5% of the accused/tried are actually innocent. The corresponding probabilities for the verdicts ‘not guilty’ and ‘not proven’ are 0.90 and 0.30, respectively.


(a) What is the probability that an accused person is actually innocent?
(b) What is the probability that an innocent person will be found guilty?

Homework Equations

The Attempt at a Solution

I know how to do Question 1.

accused = 100%
f= innocent
E1=guilty
E2=not guilty
E3=not proven...therfore maximum of 3 events possible
E1, E2, E3=Accused = 100%

Using law of total probability p(f)=p(f|E1)P(E1) + p(f|E2)P(E2)...n times

p(f)=p(f|E1)P(E1) + p(f|E2)P(E2)+ p(f|E2)P(E3)


p(f|E1)=0.05

p(f|E1)=0.9

p(f|E1)=0.3
...all other info is given also

= 0.05 × 0.65 + 0.90 × 0.25 + 0.30 × 0.10
= 0.2875


HOW DO YOU DO Question 2!

 

[vela]

You're trying now to find P(E1|f). Hint: Look up Bayes Theorem.

 

[HallsofIvy]

Suppose there were 10000 cases. Then there were 6500 "guilty" verdicts, 2500 "not guilty", and 1000 "not proven" (That's a British verdict. In the United States, a person can only be found "guilty" or "not guilty" and if found "not guilty" can never be retried for that offense).

We are told that "When the court’s verdict is ‘guilty’ 5% of the accused/tried are actually innocent." which means that 0.05(6500)= 325 innocent people are found "guilty". That "The corresponding probabilities for the verdicts ‘not guilty’ and ‘not proven’ are 0.90 and 0.30, respectively" tells us that .90(2500)= 1750 innocent people are found "not guilty" and 0.3(1000)= 300 innocent people get verdicts of "non proven".

That is, there are a total of 325+ 1750+ 300= 2375 innocent people in the 10000 cases. Unless I have made an arithmetic error (always possible) the probability that an innocent person is accused is 2375/10000= .2375, not .2875.

Of those 2375 innocent people, 325 were found "guilty", so the probability that an innocent person is found "guilty" is ____.