352InclExclHandout - event decompositions. Therefore, Hence...

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Inclusion - Exclusion Theorem: For let denote the event that exactly m among the events occur simultaneously. Then , where . (Note that the formula for contains summands.) Proof: Let be the partition of the sample space determined by the possible events of the form where each is either . For let and let such that . Then . Now consider E -event probabilities. Since the right hand side sum includes if and only if exactly of the summands in the expression for include in their E-
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Unformatted text preview: event decompositions. Therefore, Hence Observe that then Therefore, Q.E.D. Theorem: For let denote the event that at least m among the events occur simultaneously. Then Proof: Define the B-events as in the previous theorem. so the formula is valid when . Now assume the formula is valid for fixed and consider Therefore, the validity of the formula for fixed implies the validity of the formula for . Hence, the theorem is proved by induction. Q.E.D....
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This note was uploaded on 01/15/2010 for the course MATH 352 taught by Professor M.tao during the Spring '09 term at University of Victoria.

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352InclExclHandout - event decompositions. Therefore, Hence...

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