Question

Suppose that there are 5 blood types in the population, named type 1 through type 5, with probabilities p1, p2,. . . , p5. A crime was committed by two individuals. A suspect, who has blood type 1, has prior probability p of being guilty. At the crime scene blood evidence is collected, which shows that one of the criminals has type 1 and the other has type 2. Find the posterior probability that the suspect is guilty, given the evidence.

Does the evidence make it more likely or less likely that the suspect is guilty, or does this depend on the values of the parameters p, p1,. . . , p5? If it depends, give a simple criterion for when the evidence makes it more likely that the suspect is guilty.

Answer 1

P(guilty|evidence) = P(evidence|guilty)P(guilty)/(P(evidence|guilty)P(guilty)+P(evidence|not guilty)P(not guilty)).

P(guilty) = p

P(not guilty) = 1 - p

P(evidence|guilty) = 1/2 because one criminal has blood type 1

P(evidence|not guilty) = 1/5 because there are 5 blood types.

So I get P(guilty|evidence) = 0.5p/(0.5p + 0.2(1-p))