A judge is 35% sure that Mary has committed a crime. Jack and Tim are two witnesses who know whether Mary is innocent or guilty. However, Jack is Mary's friend and will lie with probability 0.25 if Mary is guilty. He tells the truth if she is innocent. Tim is Mary's enemy and will lie with probability 0.30 if Mary is innocent. He will tell the truth if Mary is guilty. What is the probability that Mary is guilty if Jack and Tim gave conflicting testimonies?
Another witness, David, convinces the judge that there is 85% chance that the criminal is left-handed. If 23% of the population is left-handed and Mary is also left-handed, with this new information, how certain should the judge be of the guilt of Mary?
By the Law of Total Probability, Let G be the event where Mary is guilty. Let C be the event where Jack and Tim gave conflicting testimonies.
P(C) = P(C|G)P(G)+P(C|G^c)P(G^c) = 0.25*0.35 + 0.30*0.65 = 0.2825.
I need help with the 2nd part of the question...
Thanks in advance!