hw2_sol - IEOR 172 Probability and Risk Analysis for...

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IEOR 172: Probability and Risk Analysis for Engineers, Fall 2007 Homework 2 Solution Chapter 3 Question 9 Denote A as urn A, and W as white ball: P( A = 2 W | 2 W ) = P( A = 2 W, 2 W ) P(2 W ) = 2 6 8 12 · 3 4 + 4 12 · 1 4 2 6 · 8 12 · 3 4 + 2 6 · 4 12 · 1 4 + 4 6 · 8 12 · 1 4 = 7 11 Question 12 Denote F as fail the exam, F i as fail the i th exam: (a) P( F c ) = 0 . 9 · 0 . 8 · 0 . 7 = 0 . 504. (b) P( F 2 | F ) = P( F | F 2 )P( F 2 ) P( F ) = 1 · 0 . 9 · . 2 1 - 0 . 504 = 0 . 3629 . Question 25 Let p denote the proportion of people in this town who are over 50. Let α 1 denote the proportion of time that a person under the age of to spends in the streets, and α 2 be the corresponding value for those over 50. Denote S as people on the street, then this method only compute the conditional probability of people over 50, P(over 50 | S ) = P(over 50 , S ) P( S ) = α 2 p α 1 (1 - p ) + α 2 p . When α 1 α 2 , this value is approximately equal to p . Question 37 Denote H as head, T as tail, F as fair coin and uF as unfair coin: (a) P( F | H ) = P( H | F )P( F ) P( H | F )P( H ) + P( H | uF )P( uF ) = 1 2 · 1 2 1 2 · 1 2 + 1 · 1 2 = 1 3 .
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