Hw2sol - ISyE 2027B Probability with Applications Fall 2010...

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ISyE 2027B Probability with Applications Fall 2010 Homework 2 Solutions Q1. (a) 10 points There are ( 7 2 ) + ( 6 2 ) + ( 9 2 ) ways of getting a pair out of the total of (7 + 6 + 9) socks. Hence the probability of getting a pair by randomly drawing two socks in the dark equals ( 7 2 ) + ( 6 2 ) + ( 9 2 ) ( 22 2 ) = 24 77 (b) 10 points There are ( 7 2 ) ways of getting a white pair. The probability of getting a white pair equals ( 7 2 ) ( 22 2 ) = 1 11 Q2. 20 points Let g, s be the outcomes that correspond to a gold or silver coin is found in any opened drawer. Ω = { ( g, g ) , ( g, s ) , ( s, g ) , ( s, s ) } , each element of which corresponds to the two coins found in the chosen cabinet. P (the other = s | the first one = s ) = P (( s, s ) |{ ( s, g ) , ( s, s ) } ) = P (( s, s )) P ( { ( s, g ) , ( s, s ) } ) = P (Cabinet C is chosen) P ( { Cabinet B is chosen and the first coin = s } ∪ { Cabinet C is chosen } ) = 1 3 1 3 × 1 2 + 1 3 = 2 3 Q3. 20 points Define the events R 1 and R 2 as: a red ball is drawn on the first and the second draw, respectively. The question asks you to compute P ( R 1 R 2 ). By conditioning on R 1 we find: P ( R 1 R 2 ) = P ( R 2 | R 1 ) · P ( R 1 ) = 3 4 × 1 2 = 3 8
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