Exercise 12. Switched-Conditionals Fallacy. If it bleeds, it leads: Nearly 60% of violent crime stories carried in themedia show a green fanged alien being handcuffed and escorted into a police car. Assume that (i) 57% of violentcriminals are green-fanged aliens (so the media coverage is unbiased); (ii) green-fanged aliens make up only18.7% of the population; and (iii) 0.16% of the population are violent criminals. A green-fanged alien is seen onthe way to the grocery store. What are the chances this alien is a violent criminal?

Conditional Probability: Let A and B be two events with P(A) > 0. Then the conditional probability of B given A, denoted by P(B | A), is defined as the probability of occurrent of the event B when it is assumed that the event A has already been occurred. It is given by the formula, Theorem of Total Probability: Let A1, A2, ... , An be n mutually exclusive and exhaustive events with P(Ai) > 0 for all i = 1, 2, ... , n. Let B be another event with P(B) > 0. Then by the theorem of total probability we have, Bayes' Theorem: Let A1, A2, ... , An be n mutually exclusive and exhaustive events with P(Ai) > 0 for all i = 1, 2, ... , n. Let B be another event with P(B) > 0. Then by Bayes' Theorem we have,Let A1 be the event that a randomly selected person is a violent criminal and A2 be the event that a randomly selected person is not a violent criminal B be the event that the randomly selected person is a green-fanged alien Clearly, A1 and A2 are mutually exclusive and exhaustive And B | A1 is the event that the randomly selected person is green-fanged alien given the person is a violent criminal B | A2 is the event that the randomly selected person is green-fanged alien given the person is not a violent criminal We are given, P(A1)=0.16%=0.0016P(A2)=1-P(A1)=1-0.0016=0.9984 and P(B|A1)=57%=0.57P(B|A2)=18.7%=0.187 Therefore, by applying Bayes' theorem, the probability that the person is a violent criminal given that the person is a green-fanged alien is Answer: The probability that this alien is a violent criminal is 0.0049.