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18_05_lec6 - 18.05 Lecture 6 Solutions to Problem Set#1 1-1...

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18.05 Lecture 6 February 16, 2005 Solutions to Problem Set #1 1-1 pg. 12 #9 B n = A i , C n = A i i = n i = n a) B n B n +1 ... B n = A n ( i = n +1 A i ) = A n B n +1 s B n +1 s B n +1 A n = B n C n C n +1 ... C n = A n C n +1 s C n = A n C n +1 s C n +1 b) s for all n n =1 B n s B n s A i for all n s some A i for i n , for all n i =1 s infinitely many events A i A i happen infinitely often. c) s C n s some C n = A i for some n, s all A i for i n n =1 i = n s all events starting at n. 1-2 pg. 18 #4 P (at least 1 fails) = 1 P (neither fail) = 1 0 . 4 = 0 . 6 1-3 pg. 18 #12 A 1 , A 2 , ... 1 A 2 , ..., B n = A c n 1 A n B 1 = A 1 , B 2 = A c 1 ...A c P ( n A i ) = n P ( B i ) splits the union into disjoint events, and covers the entire space. i =1 i =1 follows from: n i =1 A i = n B i i =1 take point (s) in n A i , s at least one s A 1 = B 1 , i =1 1 , if s A 2 , then s A c if not, s A c 1 A 2 = B 2 , if not...
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