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HW1_soln

# HW1_soln - Homework Set#1 Solutions IE 336 Spring 2011 1...

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Homework Set #1 Solutions IE 336 Spring 2011 1. Let S denote the sample space. If events B i partition the sample space, then: S = B 1 B 2 ∪ · · · ∪ B n If A is an event from the sample space, then A can also be expressed as: A = A S = A ( B 1 B 2 ∪ · · · ∪ B n ) = ( A B 1 ) ( A B 2 ) ∪ · · · ∪ ( A B n ) Using axiom 3: P ( A ) = P ( A B 1 ) + P ( A B 2 ) + · · · + P ( A B n ) (1) Recall the definition of conditional probability: P ( A B i ) = P ( B i ) P ( A | B i ) (2) Substituting (2) into (1): P ( A ) = P ( B 1 ) P ( A | B 1 ) + P ( B 2 ) P ( A | B 2 ) + · · · + P ( B n ) P ( A | B n ) P ( A ) = n X i =1 P ( B i ) P ( A | B i ) 2. (a) A= { 1, 2, 3, 4, 5, 6 } , B = { 1, 3, 5, 11, 13, 15, 17,19, 21 } , C = { 6, 14, 22 } (b) If two events are independent, then P ( A B ) = P ( A ) P ( B ). A B = { 1 , 3 , 5 } , so P ( A B ) = 3 18 = 1 6 . Checking independence: P ( A ) · P ( B ) = 6 18 · 9 18 = 1 6 = P ( A B ) Therefore, A and B are independent. 1

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(c) A C = { 6 } , so P ( A C ) = 1 18 . Checking independence: P ( A ) · P ( C ) = 6 18 · 3 18 = 1 18 = P ( A C ) Therefore, A and C are independent.
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