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hw4Sol - CS 440 Introduction to AI Homework 4 Solution Due...

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CS 440: Introduction to AI Homework 4 Solution Due: November 9 11:59PM, 2010 Your answers must be concise and clear. Explain sufficiently that we can easily determine what you understand. We will give more points for a brief interesting discussion with no answer than for a bluffing answer. Please email your solution to the TA at [email protected] 1 Probability 1. [10 pts.] Consider a joint probability distribution P ( X, Y, Z ) over vari- ables X, Y, and Z. Assume that the probability of each outcome is non- zero. Is the following equation always true? Prove its correctness or incorrectness. P ( X | Y, Z ) = P ( Y | X, Z ) P ( X | Z ) P ( Y | Z ) (Solution) The given equation is always true. RHS of the equation has the numerator P(Y,X,Z)/P(Z) and has the denominator P(Y,Z)/P(Z) by Bayes rule. The resulting fraction P(Y,X,Z)/P(Y,Z) = P(X | Y,Z) by Bayes rule, and this is equivalent to the LHS of the equation. 2. [10 pts.] After your yearly checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease and that the test is 95% accurate (i.e., the probability of testing positive when you have the disease is 0.95 and the probability of testing negative when you don’t have the disease is 0.95). The good news is that this is a rare disease, striking only 1 in 100,000 individuals. Why is it good news that

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hw4Sol - CS 440 Introduction to AI Homework 4 Solution Due...

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