Solution_Quiz1

# Solution_Quiz1 - 3 Prove that if A 1,A 2 is decreasing then...

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1 February 2, Wednesday at 1:00 p.m. Math 335-Quiz1 Name: KEY You must answer all questions. To receive full marks, you must show all your work and give a complete answer . 1. An electronic game contains three components arranged in the series-parallel circuit shown below. At any time, each component may or may not be operative, and the game will operate only if there is continuous circuit from P to Q . Let A be the event that the game will operate; let B be the event that the game will operate though component x is not operative; and let C be the event that the game will operate though component y is not operative; Using the notation in which (0 , 0 , 1), for example, denotes that component z is operative, but components x and y are not. (a) List the elements of the sample space S , A , B , and C . S = { (0 , 0 , 0) , (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) , (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) , (1 , 1 , 1) } A = { (1 , 0 , 1) , (0 , 1 , 1) , (1 , 1 , 1) } B = { (0 , 1 , 1) } C = { (1 , 0 , 1) } (b) Determine which pairs of events are mutually exclusive. 2. Prove that 0 P ( A ) 1. ∅ ⊂ A S 0 = P ( ) P ( A ) P ( S ) = 1

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Unformatted text preview: 3. Prove that if A 1 ,A 2 ,... is decreasing then P ( lim n-→∞ A n ) = lim n-→∞ P ( A n ). Since A 1 ,A 2 ,... is a sequence of decreasing events [hence A c 1 ,A c 2 ,... is a sequence of increasing events] n \ i =1 A i = lim n →∞ A n Deﬁnition of sequence of decreasing events (1) n [ i =1 A c i = lim n →∞ A c n Deﬁnition of sequence of increasing events (2) P ( lim n →∞ A c n ) = lim n →∞ P ( A c n ) Proved in the class (3) 2 Then ( n \ i =1 A i ) c = n [ i =1 A c i P [( n \ i =1 A i ) c ] = P ( n [ i =1 A c i ) P [( n \ i =1 A i ) c ] = 1-P ( n \ i =1 A i ) = P ( n [ i =1 A c i ) = P ( lim n →∞ A c n ) = lim n →∞ P ( A c n ) = lim n →∞ (1-P ( A n )) 1-P ( n \ i =1 A i ) = 1-P ( lim n →∞ A n ) = 1-lim n →∞ P ( A n ) P ( lim n →∞ A n ) = lim n →∞ P ( A n )...
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Solution_Quiz1 - 3 Prove that if A 1,A 2 is decreasing then...

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