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Therefore in this question the probability that he is innocent is 0.1743
The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a claim to be true. A number between 0 and 1 is the probability of an event, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.
Here,
In this case, let G represent the state of guilt and H represent the state of innocence.
and Let K stand for the situation where the truth serum shows that someone is guilty.
We are aware that 90% of the guilty suspects receive just verdicts, i.e. P(K|G)=0.9.
Additionally, we know that innocent suspects are mistakenly identified 1% of the time, or P(K|H)=0.01.
When a suspect is chosen from a group of suspects, of whom 5% committed a crime, and the serum suggests that the suspect is guilty, we must determine the likelihood that the suspect is innocent.
We must therefore determine the probability P(H|K).
A group of people was chosen for the suspect, of which 5% were criminals.
so the probability that selected person is guilty is
P(G) = 0.05
Therefore, the probability that selected person is not guilty is
P(H) = 1 – P(G) = 1 – 0.05 = 0.95
Then,
Using Bayes' Theorem, we get
P(H|K) =[tex]\frac{P(H)P(K|H)}{P(H)P(K|H) + P(G)P(K|G)}[/tex]
= [tex]\frac{0.95*0.01}{0.95*0.01 + 0.05*0.9}[/tex] =0.1743
Therefore in this question the probability that he is innocent is 0.1743
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