LEC03 Axioms & Add Rule NEW

LEC03 Axioms & Add Rule NEW - 4 A Venn Diagram P A =.10...

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Definition: Probability for Experiments with Discrete Sample Spaces An experiment has discrete sample space S = {e 1 , e 2 , ... }. The experiment in repeated N times. The observed outcome is e i a fraction f i of the times such that f 1 +f 2 + ... = 1. Definition: P(e i ) is the probability of event e i As N , f i P(e i )
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Three Axioms: P(E) is the probability of event E 1. 0 1 P E ( ) 2. P(S)=1 3. For two disjoint events E 1 and E 2 , E E 1 2 = ∅ , P E E P E P E ( ) ( ) ( ) 1 2 1 2 = + and P E P E ( ) ( ) = - 1 2
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Addition Rule: For any events E and F, P E F P E P F P E F ( ) ( ) ( ) ( ) = + - 10% of parts have defect A and 5% have defect B. 2% have both defects A and B. Find the probability a part has at least one defect. Given: event A = { part has defect A } P( A) = .10 event B = {part has defect B } P( B) = .05 event A B = { part has defects A AND B } P(A B) = .02 3
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Find the prob of at least one defect Given: P( A) = .10 P( B) = .05 P(A B) = .02 Find: E ={at least one defect} = A B P(A B) Apply the addition rule: P (A B) = P(A) + P(B) - P(A B) = .10 + .05 - .02 = .13
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Unformatted text preview: 4 A Venn Diagram P( A) = .10 P( B) = .05 P(A ∩ B) = .02 Find: P(A ∪ B) 0.02 A B 0.03 0.08 5 More on the example 0.02 A B 0.03 0.08 Here is wrong idea: P (A ∪ B) = P(A) + P(B) No!!! counting the intersection twice 6 Another Example A system consists of two components, A and B, in parallel • Component & System data The probability A works is 0.88 The probability B works is 0.71 The probability both work is 0.68 • Find the probability the system works Given: P(A) = 0.88 P(B) = 0.71 P(both work) = P ( ) = 0.68 Find: 7 Another example Parts can have two types of defects, A and B. 150 parts are inspected an the results are shown below: Defect B yes no Defect A yes 5 10 no 15 120 Define notation, use it, and solve. If a part is randomly selected, • Find prob of defect A • Find the prob of defect B • of BOTH Defect A and B • Find prob of No defects 8...
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