rpra2 - Engineering Risk Benefit Analysis 1.155, 2.943,...

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RPRA 2. Elements of Probability Theory 1 Engineering Risk Benefit Analysis 1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82, ESD.72, ESD.721 RPRA 2. Elements of Probability Theory George E. Apostolakis Massachusetts Institute of Technology Spring 2007
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RPRA 2. Elements of Probability Theory 2 Probability: Axiomatic Formulation The probability of an event A is a number that satisfies the following axioms (Kolmogorov): 0 P(A) 1 P(certain event) = 1 For two mutually exclusive events A and B: P(A or B) = P(A) + P(B)
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RPRA 2. Elements of Probability Theory 3 Relative-frequency interpretation Imagine a large number n of repetitions of the “experiment” of which A is a possible outcome. If A occurs k times, then its relative frequency is: It is postulated that: n k ) A ( P n k lim n
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RPRA 2. Elements of Probability Theory 4 Degree-of-belief (Bayesian) interpretation No need for “identical” trials. The concept of “likelihood” is primitive, i.e., it is meaningful to compare the likelihood of two events. P(A) < P(B) simply means that the assessor judges B to be more likely than A. Subjective probabilities must be coherent , i.e., must satisfy the mathematical theory of probability and must be consistent with the assessor’s knowledge and beliefs.
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RPRA 2. Elements of Probability Theory 5 Basic rules of probability: Negation S E Venn Diagram ___ E S E E = = = + 1 ) S ( P ) E ( P ) E ( P ) E ( P 1 ) E ( P =
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RPRA 2. Elements of Probability Theory 6 Basic rules of probability: Union () () () + + = + =+ = = ∑∑ I U K N 1 i 1 N 1 N 1 i N 1 i j j i N 1 i i N 1 i A P 1 A A P A P A P = N 1 i i N 1 i A P A P U Rare-Event Approximation:
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RPRA 2. Elements of Probability Theory 7 Union (cont’d) For two events: P(A B) = P(A) + P(B) – P(AB) For mutually exclusive events: P(A B) = P(A) + P(B)
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RPRA 2. Elements of Probability Theory 8 Example: Fair Die Sample Space: {1, 2, 3, 4, 5, 6} (discrete) “Fair”: The outcomes are equally likely (1/6). P(even) = P(2 4 6) = ½ (mutually exclusive)
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RPRA 2. Elements of Probability Theory 9 Union of minimal cut sets ∑∑ = + =+ = = + + = N 1 i i 1 N 1 N 1 i N 1 i j j i N 1 i i T M M M M X 1 ) ( ...
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rpra2 - Engineering Risk Benefit Analysis 1.155, 2.943,...

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