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Homework assignments 5 and 6(3).pdf

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Franz Huber

Homework assignment 5 (due November 7) A probability space is a triple (W, A, Pr) consisting of an arbitrary non-empty set of worlds or possibilities W, an algebra A of propositions over W, and a probability measure Pr: A→ ℜ. A probability measure Pr: A → ℜ is a function from the algebra of propositions Ainto the real numbers ℜ such that for all B and C in A: (1) Pr(B) ≥ 0. (2) Pr(W) = 1. (3) Pr(B∪C) = Pr(B) + Pr(C) if B∩C = ∅. If Pr(C) > 0, the conditional probability of B given C, Pr(B|C), is defined as Pr(B∩C)/Pr(C). If Pr(C) = 0, the conditional probability of B given C, Pr(B|C), is not defined. Show that the following claims hold true in a probability space (W, A, Pr). Claim I: For all B in A, Pr(B) = Pr(B|W). (1 point) Claim II: For all B in A, Pr(B|B) = 1 if Pr(B) > 0. (1 point) Claim III: Pr(∅) = 0. (1 point) Claim IV: For all B in A, Pr(B) = 1 – Pr(W\B). (1 point) Claim V: For all B and C in A, Pr(B) = Pr(B∩C) + Pr(B∩(W\C)). (1 point) (Hint: use (IV) from homework assignment 1) Homework assignment 6 (due November 14) Claim VI: For all B and C in A, Pr(B∪C) = Pr(B) + Pr(C) – Pr(B∩C). (1 point) (Hint: use (IV) and (VIII) from homework assignment 1) Claim VII: For all B and C in A, Pr(B∩C) = Pr(B)xPr(C|B), whe
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