pfinal

# pfinal - Philosophy 148"Practice Final Exam[This is just...

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Philosophy 148 — “Practice Final Exam” [This is just a “practice exam”, but it will be very similar in structure and content to the actual ﬁnal exam.] This exam contains ﬁve problems (worth 25 points each). You are to work four of these ﬁve problems. You also have the option of working a ﬁfth problem for extra-credit (10 points worth). If you choose to try all ﬁve, then make sure you clearly mark which one is going to count as your extra-credit problem. (If you don’t specify, we’ll simply grade the last of the ﬁve problems in your blue book as your extra credit problem.) Read the instructions for each problem carefully, as some problems will involve a choice among multiple questions. Write all answers (legibly!) in your blue book(s). You have three hours to complete the exam. You are not to consult books or notes. 1 Proving a Theorem Both Axiomatically and Algebraically Recall, the three Kolmogorov axioms for probability calculus (plus the deﬁnition of conditional probability) are: 1. For all propositions p , Pr p p q ¥ 0. 2. Pr pJq 1, where J is any tautological proposition. 3. For all propositions p and q , if p and q are mutually exclusive ( i.e. , if p q )(K , where K is any contradictory proposition), then Pr p p _ q q Pr p p Pr p q q . 4. For all propositions p and q , Pr p p | q q df Pr p p q q Pr p q q . This problem has two parts, and you are to do both of them: (a) Using these axioms (plus deﬁnition), prove that if Y Z ( X , then Pr p X | Y q ¥ Pr p Z | Y q . You may also use logic, Skyrms’s rules, or any theorems proved during the semester. Be sure to justify each step in your proof. (b) Prove the same theorem algebraically , using the following variables in your algebra. X Y Z Pr T T T a T T F b T F T c T F F d F T T e F T F f F F T g F F F h 2 Calculating Two Conﬁrmation Measures on a Given Probability Model Consider the probability model M

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pfinal - Philosophy 148"Practice Final Exam[This is just...

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