L15_InclusionExclusion

L15_InclusionExclusion - 1-1COMP170Discrete Mathematical...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 5.2, pp. 224-233Inclusion-ExclusionVersion 2.0: Last updated, May 13th, 2007Slidesc2005 by M. J. Golin and G. Trippen2-1Unions and Intersections•The Probability of a Union of Events•The Principle of Inclusion and Exclusionfor Counting•The Principle of Inclusion and Exclusionfor Probability3-1The Probability of a Union of EventsInP(E) +P(F), weights of elements ofE∩Feach appeartwice, while weights of all other ele-ments ofE∪Feach appear exactlyonce.Venn Diagram3-2The Probability of a Union of EventsInP(E) +P(F), weights of elements ofE∩Feach appeartwice, while weights of all other ele-ments ofE∪Feach appear exactlyonce.Venn Diagram3-3The Probability of a Union of EventsInP(E) +P(F), weights of elements ofE∩Feach appeartwice, while weights of all other ele-ments ofE∪Feach appear exactlyonce.Venn DiagramSample Space3-4The Probability of a Union of EventsInP(E) +P(F), weights of elements ofE∩Feach appeartwice, while weights of all other ele-ments ofE∪Feach appear exactlyonce.Venn DiagramSample SpaceEvents4-1John Vennb. 1834, d. 1923British Mathematician who continuedthe work of Boole. Although he wasnot the first person to use diagramsin formal logic, he seems to have beenthe first to formalize their usage andgeneralize them.For more, see the survey of Venn diagrams athttp://www.combinatorics.org/Surveys/ds5/VennJohnEJC.html5-11215-2P(E)+P(F)counts probability weights ofeach element ofE∩Ftwice.1215-3P(E)+P(F)counts probability weights ofeach element ofE∩Ftwice.121Thus, to get a sum that includes probability weight of eachelement ofE∪Fexactlyonce, we mustsubtractweight ofE∩FfromP(E) +P(F).5-4P(E)+P(F)counts probability weights ofeach element ofE∩Ftwice.121Thus, to get a sum that includes probability weight of eachelement ofE∪Fexactlyonce, we mustsubtractweight ofE∩FfromP(E) +P(F).P(E∪F) =P(E) +P(F)-P(E∩F)(*)6-1If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?6-2If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?EventE: Sum is evenEventF: Sum is8or more6-3If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?EventE: Sum is evenEventF: Sum is8or moreP(E) =126-4If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?P(F) =536+436+336+236+136=1536EventE: Sum is evenEventF: Sum is8or moreP(E) =126-5If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?P(F) =536+436+336+236+136=1536EventE: Sum is evenEventF: Sum is8or moreP(E) =12P(8)P(9)P(10)P(11)P(12)6-6If you roll two dice, what is the probability of eitheran even sum or a sum of8or more (or both)?...
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

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L15_InclusionExclusion - 1-1COMP170Discrete Mathematical...

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