Investigation 6: Forensic Evidence (due Monday, October 4)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones.

Bayes’ Theorem was applied by expert witnesses testifying in a rape trial in Pittsburgh in the mid 1980’s. The defendant was accused of raping seven women in the Shadyside district of the city over a period from April 18, 1985, to January 30, 1986. By analyzing body secretion evidence taken from the scenes of the crimes, a forensic expert concluded that the assailant had the blood characteristics and genetic markers of type B, secretor, PGM 2+1-. She further testified that only .32% of the male population of Allegheny County had these blood characteristics and that the defendant himself was a type B, secretor, PGM 2+1-. The natural question to ask is how a juror should update the probability of the defendant’s guilt in light of this quantitative forensic evidence.

a) Let G represent the event that the defendant is guilty, and let E represent the forensic evidence that the criminal’s blood type was type B, secretor, PGM 2+1-. Let P(G) represent the prior probability that a juror assigns to the defendant's guilt before hearing the forensic evidence. Rewrite Bayes’ Theorem terms of G and E for expressing how to find the updated probability of guilt, conditional on the forensic evidence, from the prior probability of guilt.

b) What is P(E|G) in this situation? [Hint: Remember that the defendant is a type B, secretor, PGM 2+1-, so if he is guilty, then the forensic evidence would surely have been that of someone with type B, secretor, PGM 2+1- blood.]

c) What is P(E|G') in this situation? [Hint: Assume that if the defendant did not commit the crimes, then some other “random” male in Allegheny County did. Note that this is not the “complement” of P(E|G).]

d) Use your answers to the preceding questions to express the updated probability of guilt P(G|E) as a function of the prior probability of guilt P(G).

e) Calculate the updated probabilities of guilt P(G|E) for the following prior probabilities P(G): .5, .25, .1, .01, .001, and .00000278.

The last entry in this list deserves special mention. The defense in this case argued that the prior probability of guilt should be 1 in 360,000, the estimated number of males in the appropriate age group in Allegheny County. The updated probability of guilt then becomes just 1 in 1150, the number of males with the same blood characteristics in the appropriate age group in Allegheny County.

f) Construct a graph of P(G|E) as a function of P(G), for values of P(G) ranging from 0 to .5. [Hint: Feel free to use Excel or another software package.]

g) Determine the smallest prior probability P(G) for which the updated probability of guilt P(G|E) exceeds .99.

h) Now suppose that in a different trial, the .0032 value is different; call it p. If a juror’s prior probability of guilt is .2, what values of p would produce an updated probability of guilt that exceeds .99? [Show the details of your calculation.]