Question

In a trial, the judge is 65% sure that Susan has committed a crime. Julie and Robert are two witnesses who know whether Susan is innocent or guilty. However, Robert is Susan’s friend and will lie with probability 0.25 if Susan is guilty. He will tell the truth if she is innocent. Julie is Susan’s enemy and will lie with probability 0.30 if Susan is innocent. She will tell the truth if Susan is guilty. What is the probability that Susan is guilty if Robert and Julie give conflicting testimony?

Answer 1

Probability of Susan committing crime according to judge = 0.65

Probability of Susan not committing crime according to judge = 1- 0.65 =0.35

If Susan is guilty:

Robert will lie will probability 0.25 and and Julie will tell truth with probability 1.

Probability of conflicting testimony if Susan is guilty P(G and C)= 0.65 * 0.25 * 1 = 0.1625

If Susan is not guilty:

Robert will tell truth will probability 1 and and Julie will lie with probability 0.30.

Probability of conflicting testimony if Susan is not guilty, P(G' and C) = 0.35 * 1 * 0.30 = 0.105

Probability of conflicting testimony, P(C) = P(G and C) + P(G' and C)

= 0.1625 + 0.105 = 0.2675

Probability that Susan is guilty given that Robert and Julie give conflicting testimony, P(G|C) = P(G and C)/ P(C)

= 0.1625 / 0.2675

= 0.6074