ECE313.Lecture12

ECE313.Lecture12 - ECE 313 Probability with Engineering...

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Conditional Probability Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 41 Reminders Axiomatic theory of probability The set of all possible outcomes of an experiment is the sample space Events are subsets of An event is said to have occurred if the outcome of the experiment is a member of the event (that is, subset of ) A and A c are a partition of On every trial, one of A and A c must occur
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 41 20-20 hindsight is a wonderful thing Once the experiment has been performed and the outcome is known, we have perfect and complete knowledge For each pair of events A and A c , we can tell which one occurred and which one didn’t — just check which set A or A c the observed outcome belongs to! There is no probabilistic consideration any more, and we do not need to think about the chances of A (or A c ) occurring
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 41 A little learning is a dangerous thing. . Now suppose that the experiment has been performed, but we do not know the outcome exactly All we know is that the outcome is some member of the event A, but we do not know which member of A it is Put another way, we are told that the event A has occurred, but nothing else To avoid trivial cases, assume that A is not a singleton or elementary event
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 41 … B or ¬ B? That is the question… The experiment has been performed and we know that the event A has occurred, that is, the outcome is some member of A Did the event B occur? or did B c occur? Unlike the case of perfect knowledge, we cannot tell whether B or B c occurred AB and AB c are a partition of A If the outcome AB, then B occurred: if the outcome AB c , then B c occurred
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 41 The exorcism did not work… The experiment has been performed and we know that the event A occurred, that is, the outcome is some member of A We cannot tell for sure whether B occurred We have not exorcised the probability from the problem as yet — A probability question still continues to plague us Question: What are the chances that B occurred? in view of the new knowledge
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ECE 313 - Lecture 12 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 41
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ECE313.Lecture12 - ECE 313 Probability with Engineering...

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