chap4slidespt1

chap4slidespt1 - n-D Continuous Probability Spaces Uniform...

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n -D Continuous Probability Spaces Uniform probability space (0 , 1) (1 , 0) (1 , 1) x = 2 y M ( S ) = 1 M ( A ) = 1 4 P ( A ) = M ( A ) M ( S ) = 1 4 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 176
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Chapter 4 Conditional Probability and Independence ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 1
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Conditional Probability Introduction In a random experiment: knowing that a certain event B has occured may completely change the likelihood we associate to another event A . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 2
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Conditional Probability Example Experiment: Roll two fair dice, S = { ( x,y ) : ∈ { 1 , 2 ,..., 6 }} (1 , 1) (1 , 2) (1 , 3) (1 , 4) (1 , 5) (1 , 6) (2 , 1) (2 , 2) (2 , 3) (2 , 4) (2 , 5) (2 , 6) (3 , 1) (3 , 2) (3 , 3) (3 , 4) (3 , 5) (3 , 6) (4 , 1) (4 , 2) (4 , 3) (4 , 4) (4 , 5) (4 , 6) (5 , 1) (5 , 2) (5 , 3) (5 , 4) (5 , 5) (5 , 6) (6 , 1) (6 , 2) (6 , 3) (6 , 4) (6 , 5) (6 , 6) A B A : “ x + y = 11 P ( A ) = 2 / 36 If we know that B : “ x = 1 ” has occured P ( A ) = 0 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 3
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Conditional Probability Introduction Question How can we quantify our certainty that an event will happen, when: We know that another event has occurred, or that certain a priori knowledge is available? ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 4
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Conditional Probability Notation Consider a random experiment. Let A and B denote two events. Probability of A : P ( A ) . If we know that B has occured: P ( A ) P ( A | B ) Conditional Probability of A given B ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 5
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Conditional Probability Relative frequency interpretation Consider a random experiment ( S, F ,P ) . Events A , B ∈ F with P ( B ) > 0 . Experiment is repeated n times. For n large: P ( A ) η ( A ) n , P ( B ) η ( B ) n , P ( A B ) η ( A B ) n η ( · ) : number of occurrences of an event within the n repetitions. ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 6
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Conditional Probability Relative frequency interpretation Conditional relative frequency: P ( A | B ) η ( A B ) η ( B ) , provided η ( B ) is large. We have P ( A | B ) = η ( A B ) η ( B ) = η ( A B ) /n η ( B ) /n P ( A B ) P ( B ) ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 7
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Conditional Probability Definition Consider a random experiment ( S, F ,P ) and let B ∈ F and assume that P ( B ) > 0 . For every A ∈ F , the conditional probability of A given B , denoted P ( A | B ) , is defined as P ( A | B ) = P ( A B ) P ( B ) ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 8
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Conditional Probability
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chap4slidespt1 - n-D Continuous Probability Spaces Uniform...

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