hw1 - solutions - f08

# hw1 - solutions - f08 - IE 336: Operations Research -...

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IE 336: Operations Research - Stochastic Models Homework Set #1 Solutions Fall 2008 1. (Hint: Try considering A ( B 1 B 2 ∪ ··· ∪ B n ) = ( A B 1 ) ( A B 2 ) ∪ ··· ∪ ( A B n ) . ) Figure 1: Venn Diagram for 4 B events Based on the Venn Diagram we can say that A = ( A B 1 ) ( A B 2 ) ∪ ··· ∪ ( A B n ) . (1) Taking the probability we have, P ( A ) = P ( A B 1 ) P ( A B 2 ) ∪ ··· ∪ P ( A B n ); (2) by the third probability axiom, P ( A ) = P ( A B 1 ) + P ( A B 2 ) + ··· + P ( A B n ) . (3) Using the deﬁnition of conditional probability, P ( A B i ) = P ( B i ) P ( A | B i ) . (4) So, subsituting (4) into (3) we obtain, P ( A ) = P ( B 1 ) P ( A | B 1 ) + P ( B 2 ) P ( A | B 2 ) + ··· + P ( B n ) P ( A | B n ) (5) P ( A ) = n X i =1 P ( B i ) P ( A | B i ) . ( Done ) (6) 1

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2. (a) B = { 1,3 } , C = { 3,4 } (b) If two events are independent then P ( A B ) = P ( A ) P ( B ) . Event (
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## This note was uploaded on 03/29/2010 for the course IE 383 taught by Professor Leyla,o during the Spring '08 term at Purdue University-West Lafayette.

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hw1 - solutions - f08 - IE 336: Operations Research -...

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