# lect4 - INCLUSION-EXCLUSION FOR n EVENTS CONDITIONAL...

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INCLUSION-EXCLUSION FOR n EVENTS Outline INCLUSION-EXCLUSION FOR n EVENTS CONDITIONAL PROBABILITY 1 / 12 Xinghua Zheng Lect 4: More Rules of Probability; Conditional Probability

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INCLUSION-EXCLUSION FOR n EVENTS INCLUSION-EXCLUSION FOR 3 EVENTS Recall that the inclusion-exclusion rule for two events A and B says that P [ A B ] = P [ A ] + P [ B ] - P [ A B ] . What about 3 events A , B , and C ? P [ A B C ] = P [ A ] + P [ B ] + P ( C ) - ( + + ) + . 2 / 12 Xinghua Zheng Lect 4: More Rules of Probability; Conditional Probability
INCLUSION-EXCLUSION FOR n EVENTS INCLUSION-EXCLUSION FOR n EVENTS What about n events A 1 , A 2 ,..., A n ? p := P [ n i = 1 A i ] = S 1 - S 2 + S 3 - S 4 + ... + ( - 1 ) n - 1 S n , ( * ) S 1 = n X i = 1 P [ A i ] , S 2 = X 1 i < j n P [ A i A j ] , S 3 = X 1 i < j < k n P [ A i A j A k ] , . . . Moreover, the sum on the RHS of (*) is an enveloping sum : 1. If k is an odd number, then p S 1 - S 2 + S k ; e.g., p S 1 ; 2. If k is an even number, then p S 1 - S 2 - S k ; e.g., p S 1 - S 2 . 3 / 12 Xinghua Zheng Lect 4: More Rules of Probability; Conditional Probability

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INCLUSION-EXCLUSION FOR n EVENTS EXAMPLE: BRIDGE In the game of bridge, each of the 4 players North, South, East, and West receives 13 cards from an ordinary 52 card deck. A bridge hand is said to be void of some suit (e.g., spades) if it contains no cards of that suit. Assuming the deck is thoroughly shufﬂed, what is the chance p that North’s hand is void in at least one suit?
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lect4 - INCLUSION-EXCLUSION FOR n EVENTS CONDITIONAL...

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