Prelim I Solutions

Prelim I Solutions - ENGRD 2700 Spring'09 Prelim 1...

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ENGRD 2700, Spring’09 Prelim 1 Solutions Prelim 1 Solutions Feb 19, 2009 Solution to Problem 1. Let us define the events D := The building gets an award for its design E := The building gets an award for environmental efficiency We are given that P ( D ) = 0 . 16 P ( E ) = 0 . 24 P ( D E ) = 0 . 11 (a) The required probability is P ( D E ) = P ( D ) + P ( E ) - P ( D E ) = 0 . 16 + 0 . 24 - 0 . 11 = 0 . 29 (b) The required probability is P (( D E c ) ( E D c )) = P ( D E c ) + P ( E D c ) = { P ( D ) - P ( D E ) } + { P ( E ) - P ( D E ) } = P ( D ) + P ( E ) - 2 P ( D E ) = 0 . 16 + 0 . 24 - 2 × 0 . 11 = 0 . 18 (c) The required probability is P ( D c E c ) = 1 - P ( D E ) = 1 - . 29 = 0 . 71 , where the first equality follows from the observation that D c E c is the complement of ( D E ). (d) Since P ( D E ) = 0 . 11 6 = ( . 24)( . 16) = 0 . 0384 = P ( D ) P ( E ) , it follows that D and E are not independent. (e) The given statement is FALSE . Consider the example of tossing a fair coin twice. Then the sample space is Ω := { HH, HT, TH, TT } 1
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ENGRD 2700, Spring’09 Prelim 1 Solutions with all the four outcomes equally likely. Define the events A = { HH, HT } B = { HH, TH } Then P ( A ) = P ( B ) = 1 / 2. Also, P ( A B ) = P ( { HH } ) = 1 / 4 Thus, P
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