Suppose you are the detective in charge of determining the identity of the perso

Question

Suppose you are the detective in charge of determining the identity of the person who committed a murder. Suppose you know for certain that the guilty culprit is exactly one of three suspects, whom we will call suspects A, B, and C. Based on your prior experience with these individuals, but before processing forensic evidence from the crime scene, you believe the probabilities of guilt for the three suspects are P(A) = 0.8, P(B) = 0.1, and P(C) = 0.1. Then, a partial fingerprint F is found at the crime scene, and the forensic expert gives the following likelihoods for fingerprint to be from the three suspects: P(F|A) = 0.05, P(F|B) = 0.65, and P(F|C) = 0.30.

(a) What is the posterior probability of guilt for suspect A (i.e., what is the probability that A committed the murder after taking into account the partial fingerprint)? Answer to four significant figures.

(b) What is the posterior probability of guilt for suspect B? Answer to four significant figures.

(c) What is the posterior probability of guilt for suspect C? Answer to four significant figures.

Explanation / Answer

First of all we need to find P(F)=P(F|A)*P(A)+P(F|B)*P(B)+P(F|C)*P(C)=0.05*0.8+0.65*0.1+0.30*0.1=0.04+0.065+0.03=0.135

a) P(A|F)=P(F|A)*P(A)/P(F)=0.05*0.8/0.135=0.296296=0.3 approx

b) P(B|F)=P(F|B)*P(B)/P(F)=0.65*0.1/0.135=0.4814=0.48 approx

c) P(C|F)=P(F|C)*P(C)/P(F)=0.30*0.1/0.135=0.22222=0.22 approx

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