Consider a student who has the choice to buy or not buy a textbook for a course. We’ll model this as a decision problem with one Boolean decision node, $B$, indicating whether the agent chooses to buy the book, and two Boolean chance nodes, $M$, indicating whether the student has mastered the material in the book, and $P$, indicating whether the student passes the course. Of course, there is also a utility node, $U$. A certain student, Sam, has an additive utility function: 0 for not buying the book and -$100 for buying it; and $2000 for passing the course and 0 for not passing. Sam’s conditional probability estimates are as follows: \(\begin{array}{ll} P(p|b,m) = 0.9 & P(m|b) = 0.9 \\ P(p|b, \lnot m) = 0.5 & P(m|\lnot b) = 0.7 \\ P(p|\lnot b, m) = 0.8 & \\ P(p|\lnot b, \lnot m) = 0.3 & \\ \end{array}\)
You might think that $P$ would be independent of $B$ given $M$, But this course has an open-book final—so having the book helps.
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Draw the decision network for this problem.
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Compute the expected utility of buying the book and of not buying it.
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What should Sam do?