To battle against spam, Bob installs two anti-spam programs. An email arrives, which is

Chapter 2, Problem 26

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To battle against spam, Bob installs two anti-spam programs. An email arrives, which is either legitimate (event L) or spam (event Lc), and which program j marks as legitimate (event Mj ) or marks as spam (event Mc j ) for j 2 {1, 2}. Assume that 10% of Bobs email is legitimate and that the two programs are each 90% accurate in the sense that P(Mj |L) = P(Mc j |Lc)=9/10. Also assume that given whether an email is spam, the two programs outputs are conditionally independent. (a) Find the probability that the email is legitimate, given that the 1st program marks it as legitimate (simplify). (b) Find the probability that the email is legitimate, given that both programs mark it as legitimate (simplify). (c) Bob runs the 1st program and M1 occurs. He updates his probabilities and then runs the 2nd program. Let P(A) = P(A|M1) be the updated probability function after running the 1st program. Explain briefly in words whether or not P(L|M2) = P(L|M1\ M2): is conditioning on M1\M2 in one step equivalent to first conditioning on M1, then updating probabilities, and then conditioning on M2?