L15_InclusionExclusion_print

L15_InclusionExclusion_print - COMP170 Discrete...

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1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 5.2, pp. 224-233 Inclusion-Exclusion Version 2.0: Last updated, May 13th, 2007 Slides c ± 2005 by M. J. Golin and G. Trippen
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2 Unions and Intersections The Probability of a Union of Events The Principle of Inclusion and Exclusion for Counting The Principle of Inclusion and Exclusion for Probability
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3 The Probability of a Union of Events In P ( E ) + P ( F ) , weights of elements of E F each appear twice , while weights of all other ele- ments of E F each appear exactly once . Venn Diagram Sample Space Events
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4 John Venn b. 1834, d. 1923 British Mathematician who continued the work of Boole. Although he was not the first person to use diagrams in formal logic, he seems to have been the first to formalize their usage and generalize them. For more, see the survey of Venn diagrams at http://www.combinatorics.org/Surveys/ds5/VennJohnEJC.html
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5 P ( E ) + P ( F ) counts probability weights of each element of E F twice . 1 2 1 Thus, to get a sum that includes probability weight of each element of E F exactly once , we must subtract weight of E F from P ( E ) + P ( F ) . P ( E F ) = P ( E ) + P ( F ) - P ( E F ) (*)
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6 If you roll two dice, what is the probability of either an even sum or a sum of 8 or more (or both)? P ( F ) = 5 36 + 4 36 + 3 36 + 2 36 + 1 36 = 15 36 Probability of even sum of 8 or more is P ( E F ) = 5 36 + 3 36 + 1 36 = 9 36 . P ( E F ) = P ( E ) + P ( F ) - P ( E F ) = 1 2 + 15 36 - 9 36 = 2 3 Event E : Sum is even Event F : Sum is 8 or more P ( E ) = 1 2 P (8) P (9) P (10) P (11) P (12) P (8) P (10) P (12)
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7 The Union of Three events: E F G When adding P ( E ) + P ( F ) + P ( G ) , weights of elements in regions E F , F G , and E G but not E F G , are counted exactly twice but weights of elements in E F G , are counted exactly three times
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8 Subtracting weights of elements of each E F , F G , and E G doesn’t quite work, since this subtracts weights of elements in EF , FG , and EG once (good) but also subtracts weights of elements in EFG three times (bad) . So, add weights of elements in E F G back into our sum. P ( E F G ) = P ( E ) + P ( F ) + P ( G ) - P ( E F ) - P ( E G ) - P ( F G ) + P ( E F G ) . Want to calculate P ( E F G ) . Start with P ( E )+ P ( F )+ P ( G ) . This Double counts events in EF , EG , FG Triple counts events in EFG
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9 Unions and Intersections The Probability of a Union of Events The Principle of Inclusion and Exclusion for Counting The Principle of Inclusion and Exclusion for Probability
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10 Principle of Inclusion and Exclusion for Probability We now guess the general formula: P ± n [ i =1 E i ! = n
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L15_InclusionExclusion_print - COMP170 Discrete...

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