Joe frequently gets into trouble at school, and past experience shows that 80% of the time he is guilty of the offense he is accused of. Joe has just gotten into trouble again, and two other students, Chris and Dana, have been called into the principal's office to testify about the incident. Chris is Joe's friend and will definitely tell the truth if Joe is innocent, but he will lie with probability 0.2 if Joe is guilty. Dana does not like Joe and so will definitely tell the truth if Joe is guilty, but will lie with probability 0.3 if Joe is innocent.
a. What is the probability that Chris and Dana give conflicting testimonies?
b. What is the probability that Joe is guilty, given that Chris and Dana give conflicting testimonies?