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chap04 - Chapter 4 Conditional Probability and Independence...

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Unformatted text preview: Chapter 4 Conditional Probability and Independence • In the context of a random experiment, knowing that a certain event B has occured may completely change the likelihood we associate to another event A . • For example, suppose we roll two fair dice:- The sample space is S = { ( x, y ) : x, y ∈ { 1 , 2 , ..., 6 }} .- Let A denote the event that the sum x + y = 11, i.e., A = { (5 , 6) , (6 , 5) } , and let B denote the event that x = 1, i.e. B = { (1 , 1) , (1 , 2) , ..., (1 , 6) } .- Assuming that the dice are fair, the probability of A is P ( A ) = 2 / 36.- Now, suppose we know that B occurred, i.e. the first die shows 1.- Under this “condition”, event A is impossible, and its likelihood or probability becomes 0. 83 84 • Conditional probabilities provide quantitative measures of likelihood (probability) under the assumption that certain events have occurred, or equivalently, that certain a priori knowledge is available. • In certain situations, knowing that B has occurred does not change the likelihood of A ; this idea is formalized via the mathematical concept of independence. • The concepts of conditional probability and independence play a ma- jor role in the design and analysis of modern information processing systems, such as digital radio receivers, speech recognition systems, file compression algorithms, etc. c 2003 Benoˆ ıt Champagne Compiled February 2, 2012 4.1 Conditional probability 85 4.1 Conditional probability Relative frequency interpretation: • Consider a random experiment. Let A and B denote two events of interest with P ( B ) > 0. • Suppose this experiment is repeated a large number of times, say n . According to the relative frequency interpretation of probability, we have P ( A ) ≈ η ( A ) n , P ( B ) ≈ η ( B ) n , P ( A ∩ B ) ≈ η ( A ∩ B ) n (4.1) where η ( A ), η ( B ) and η ( A ∩ B ) denote the number of occurrences of events A , B and A ∩ B within the n repetitions. • Provided η ( B ) is large, the probability of A , knowing or given that B has occurred, might be evaluated as the ratio P ( A given B ) = η ( A ∩ B ) η ( B ) , (4.2) also known as a conditional relative frequency . • Using this approach, we have P ( A given B ) = η ( A ∩ B ) η ( B ) = η ( A ∩ B ) /n η ( B ) /n ≈ P ( A ∩ B ) P ( B ) (4.3) • This and other considerations lead to the following definition. c 2003 Benoˆ ıt Champagne Compiled February 2, 2012 4.1 Conditional probability 86 Definition: Consider a random experiment ( S, F , P ). 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 ) (4.4) Remarks: • This definition extends the above concept of conditional relative fre- quency to the axiomatic probability framework....
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chap04 - Chapter 4 Conditional Probability and Independence...

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