# HW1b(1) - Homework 1b 1.5.1 Show that the events Ai i I are...

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Homework 1b 1.5.1 Show that the events A i , i I are independent iff P ( B i 1 . . . B i k ) = P ( B i 1 ) . . . P ( B i k ) for all distinct i 1 , . . . , i k , where each B i r may be either A i r or A c i r . - P ( A i i m +1 ) P ( A i m +2 ) . . . P ( A i k ) -
- P ( A i 1 ) P ( A c i 2 ) . . . P ( A c i m +1 ) P ( A m +2 ) . . . P ( A i k ) = (1 - P ( A i 1 )) P ( A c i 2 ) . . . P ( A c i m +1 ) P ( A m +2 ) . . . P ( A i k ) = = P ( A c i 1 ) P ( A c i 2 ) . . . P ( A c i m +1 ) P ( A m +2 ) . . . P ( A i k ) where the third equality holds by inductive assumtions (in each intersection there are no more than m complements). Therefore the claim holds by induction. 1.5.3