A truth serum given to a suspect is known to be 94 percent reliable when the per

Question

A truth serum given to a suspect is known to be 94 percent reliable when the person is guilty and 97 percent reliable when the person is innocent. In other words, 6 percent of the guilty are judged innocent by the serum and 3 percent of the innocent are judged guilty. If the suspect was selected from a group of suspects of which only 7 percent are guilty of having committed a crime, and the serum indicates that the suspect is guilty of having committed a crime, what is the probability that the suspect is innocent?

Explanation / Answer

P(Judged Guilty | Guilty) = 0.94, so that means that
P(Judged Innocent | Guilty) = 0.06

P(Judged Innocent | Innocent) = 0.97 so that means that
P(Judged Guilty | Innocent) = 0.03.

P(Guilty) = 0.07, which means that
P(Innocent) = 0.93.

We need to find P(Innocent | Judged Guilty).

by using bayes rule we have

P(A|B) and you want to find P(B|A). Here we know P(Judged Guilty | Innocent), and we want to find P(Innocent | Judged Guilty).

P(Innocent | Judged Guilty) =

P(Innocent)P(Judged Guilty | Innocent)
--------------------------------------...
{P(Innocent)P(Judged Guilty | Innocent) + P(Guilty)P(Judged guilty | Guilty)}

= {0.97*0.03} / {0.97*0.03 + 0.07*0.94}

= 0.0291 / 0.0949

=0.3066

30.66%

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