math319_S10_WS4

math319_S10_WS4 - own hat(This is the expression that...

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Probability and Statistics – Math319, Spring 2010 Worksheet 4 February 18, 2010 The Matching Problem At the end of their graduation ceremony, N students throw up their hats in the air to celebrate. When the hats land on the ground, they are completely mixed up. Each graduate randomly selects a hat from the collection on the ground. What is the probability that none of the graduates selects her/his own hat? Note: we’ll make use of the following formula below. For n events E 1 ,E 2 ,...,E N , P ( E 1 E 2 ∪ ··· ∪ E N ) = N X i =1 P ( E i ) - X 1 i 1 <i 2 N P ( E i 1 E i 2 ) + ··· + ( - 1) r +1 X 1 i 1 <i 2 < ··· <i r N P ( E i 1 E i 2 ··· E i r ) + ··· + ( - 1) N +1 P ( E 1 E 2 ··· E N ) = N X r =1 ( - 1) r +1 X 1 i 1 <i 2 < ··· <i r N P ( E i 1 E i 2 ··· E i r ) . ( ) 1. Let E i , i = 1 , 2 ,...,N, be the event that the i -th graduate selects her/his own hat. Give an expression for the probability that at least one of the graduates selects her/his own hat. Then relate your answer to identity ( ) above. 2. Now relate your answer from 1 to an expression for the probability that none of the graduates selects her/his
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Unformatted text preview: own hat. (This is the expression that we’ll ultimately be interested in.) 3. Let’s present the outcome of the experiment in the form of an N-vector ( x 1 ,x 2 ,...,x i ,...,x N ) , where x i , i = 1 ,...,N, is the hat selected by the i-th graduate. How many possible outcomes of the experiment are there? (By assumption, all outcomes are equally likely.) 4. The expression P ( E i 1 E i 2 ··· E i r ) appears in identity ( † ). Give a formula for it. What does it represent in this example? 5. Use your answer to 4 to complete the following X 1 ≤ i 1 <i 2 < ··· <i r ≤ N P ( E i 1 E i 2 ··· E i r ) = 6. Now write down an expression for P N [ i =1 E i ! = 7. Finally, using your answer to 6, write down an expression for the probability that none of the graduates selects her/his own hat. 2...
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This note was uploaded on 10/14/2010 for the course MATH 319 taught by Professor Clionagolden during the Spring '10 term at Bard College.

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math319_S10_WS4 - own hat(This is the expression that...

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