Calculating Probabilities in Criminal Verdicts: Innocence and Guilt Explained

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Homework Statement

A 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!

 

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You need to find what P(E_1|g) is. However, you only know what P(g|E_1) is. In this situation, the only thing which you can do is applying Bayes theorem...