Question: Bayess Rule You are concerned that a member of your family may be doing something illegal, e g., manufacturing narcotics, that theyre not telling you about. Then one day you discover that this family member has a second phone theyve been hiding from you.

a. If they are involved in the drugs trade, you are confident from watching TV that these folks almost always (.92 probability) have a separate phone for their drug operations. Suggest at least 2 reasons that its possible (.04 probability) that he/she would have a second phone but not be making drugs.

b. Based on what else you know about this person, you were already a little (.01 probability) suspicious. Calculate the probability that any one individual would have a second phone, based on the given information.

c. Use Bayess Rule to calculate the posterior probability of this relative being a drug manufacturer, after you find the second phone. E.g., what is Pr(drugs|2phones.)? Whats your posterior if you do not find a second phone?

d. Is this a convincing signal that your relative is manufacturing drugs? Briefly explain. Would it be more or less (or equally) convincing if the person in question seemed like the kind that would be likely to have an extramarital affair? To be concrete, you may want to attempt the calculation again with Pr(2phones|nodrugs) = 0.5.

e. How would your estimate change if you had a stronger prior, e g., you know your relative has a working knowledge of chemistry and access to lab equipment for making drugs? To be concrete, assume that this raises your initial suspicion (from a probability of .01 to .25). Could this explain a scenario in which the person in question actually tells you they have a bag containing $500,000 of drug money and you still dont believe them?