math319_S10_Worksheet5

# math319_S10_Worksheet5 - , i = 1 ,...k, to be the event...

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Probability and Statistics – Math319, Spring 2010 Worksheet 5 March 4, 2010 The Matching Problem (continued) What is the probability that exactly k graduates each pick their own hat (and each of the remaining do not pick their own hat)? Recall: from before, we have that P (none of graduates picks own hat) = N X i =0 ( - 1) i i ! . ( * ) 1. First, let’s ﬁx a particular set of k graduates. (There will be ways of choosing this set.) Let E be the event that each of the k graduates in this group selects her/his own hat. Let F be the event that none of the N - k remaining selects her/his own hat. As a ﬁrst step, for this ﬁxed set of k , we would like to calculate the desired probability = , using conditional probability. 2. In order to calculate P ( E ), we deﬁne G i

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Unformatted text preview: , i = 1 ,...k, to be the event that the i-th member of the ﬁxed set picks her/his own hat. Then, describing event E in terms of the G i , i = 1 ,...k , we have that P ( E ) = = , using the multiplication rule, = , evaluating each term, = . 3. Next, we want to calculate P ( F | E ), or in words, P ( F | E ) = P ( ) = , using ( * ), since the remaining ( N-k ) graduates are choosing from among their own ( N-k ) hats. 4. So, for the ﬁxed set of k graduates, the probability that we wish to calculate is P ( EF ) = . 5. Accounting for the number of ways we could have chosen the ﬁxed set of k graduates, this gives us the overall answer of P (exactly k matches) = , = ....
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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_Worksheet5 - , i = 1 ,...k, to be the event...

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