Practice exam question #3 for Bayesian Decision

Question 3 Suppose a crime has been committed and that the criminal has le some physical evidence at the scene that when analyzed in a laboratory contains \"chemical X". Suppose that chemical X is pmscnt in 1999 of 2000 knoWn poisons. A suspect let's call him Bob, who is found to have traces of chemical X on him is put on trial. In court the prosecutor argues as follows \"The chances that a guilty person has used a poison containing chemical X is 1999 in 2000. Bob has traces of chemical X. Therefore, the chances that Bob is innocent is just 1 in 2000.\" Assume that there are only 51000 people who could have committed the crime and that ifBoh was innocent the probability of detecting chemical X on their person would he 1 in 2, because chemical X can be found in household items. a) State whether you agree with the prosecutor's argtnnent and explanr your reasoning. b} What is the prior probability of Bob being guilty? c) What is the likelihood of Bob haying traces of chemical X if he was guilty? d} What is the likelihood of Bob haying traces of chemical X if he was innocent? e) Use Bayes Theorem to calculate the probability of Bob being guilty given he has traces of chemical X on him. f) Using the answer from Question 3 e} state whether the prosecutor was correct in his reasoning