# Let A , B and C be arbitrary events in the same sample...

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Homework 1a 1.2.2 Let A , B and C be arbitrary events in the same sample space. Let D 1 be the event that at least two of the events A , B , C occur; D 2 =exactly two of the events A , , occur 3 =at least of the events A , , occur 4 =exactly one of the events A , , occur 5 =not more than two of the events A , , occur Each of the events 2 through D 3 can be expressed in terms of A , , and by using unions, intersections, and complements. Solution. D 1 = ( A B ) ( A C ) ( B C ) D 2 = ( A B C c ) ( A B c C ) ( A c B C ) D 3 = A B C D 4 = ( A B c C c ) ( A c B C c ) ( A c B c C ) D 5 = A c B c C c 1.2.5 Establish the following: ( a, b ) = [ n =1 ( a, b - 1 n ] = [ n =1 [ a + 1 n , b ) [ a, b ] = \ n =1 [ a, b + 1 n ) = \ n =1 ( a - 1 n , b ] Solution. If x ( a, b ), a < x < b which implies b - x > 0. There exists an N N such that 1 N < b - x , which implies a < x < b - 1 N , and x [ n =1 ( a, b - 1 n ]. So, ( a, b ) [ n =1 ( a, b - 1 n ]. If x [ n =1 ( a, b - 1 n ], then x ( a, b - 1 N ] for some N N . Since 1 N > 0, we know a < x b - 1 N < b , so x ( a, b ). Thus, ( a, b ) = [ n =1 ( a, b - 1 n ]. B C D B C D B C D B C D B C