Question

(a) Your initial belief is that a defendant in a court case is guilty with probability 0.5. A witness comes forward claiming he saw the defendant commit the crime. You know the witness is not totally reliable and tells the truth with probability p. Use Bayes’ theorem to calculate the posterior probability that the defendant is guilty, based on the witness’s evidence.

(b) A second witness, equally unreliable, comes forward and claims she saw the defendant commit the crime. Assuming the witnesses are not colluding, what is your posterior probability of guilt?

(c) If p 0.5, compare the answers to (a) and (b). How do you account for this curious result?

Answer 1

A)

G = defendant is guilty

NG = defendant is not guilty

WCG = witness claims defendant is guilty

WCNG = witness claims defendant is not guilty

G = defendant is guilty

NG = defendant is not guilty

WCG = witness claims defendant is guilty

WCNG = witness claims defendant is not guilty

Witness tells the truth when

1. WCG|G
2. WCNG|NG

P(G) = 0.5

P(NG) = 0.5

From a)

P(WCG|G) = p

P(WCNG|G) = 1-p

From b)

P(WCNG|NG) = p

P(WCG|NG) = 1-p

We want to know,

P(G|WCG) = P (WCG and G) / P(WCG) = {P(WCG|G) * P(G)} / {P(WCG|G) * P(G) + P(WCG|NG) * (NG)} = {p*0.5} / {p+1-p} * 0.5 = p