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Difference between revisions of "Montmort matching problem"

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''derangement problem''
 
''derangement problem''
  
Take two sets, $A$ and $B$, and a [[bijection]], $\phi$, between them. (E.g., take $n$ married couples and let $A$ be the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at least one  "correct match"  (i.e. coincides with $\phi$ in at least one element). Asymptotically, this [[probability]] is $1-e^{-1}$. This follows immediately from the formula given in [[Classical combinatorial problems|Classical combinatorial problems]] for the number of permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$.
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Take two sets, $A$ and $B$, and a [[bijection]], $\phi$, between them. (E.g., take $n$ married couples and let $A$ be the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at least one  "correct match"  (i.e. coincides with $\phi$ in at least one element). Asymptotically, this [[probability]] is $1-e^{-1}$. This follows immediately from the formula given in [[Classical combinatorial problems|Classical combinatorial problems]] for the number of ''[[derangement]]s'': permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$.
  
 
This problem was considered first by P.R. de Montmort (around 1700) in connection with a card game known as the  "jeu du treize",  "jeu de rencontre"  or simply  "rencontre".
 
This problem was considered first by P.R. de Montmort (around 1700) in connection with a card game known as the  "jeu du treize",  "jeu de rencontre"  or simply  "rencontre".

Revision as of 07:24, 2 December 2016

derangement problem

Take two sets, $A$ and $B$, and a bijection, $\phi$, between them. (E.g., take $n$ married couples and let $A$ be the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at least one "correct match" (i.e. coincides with $\phi$ in at least one element). Asymptotically, this probability is $1-e^{-1}$. This follows immediately from the formula given in Classical combinatorial problems for the number of derangements: permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$.

This problem was considered first by P.R. de Montmort (around 1700) in connection with a card game known as the "jeu du treize", "jeu de rencontre" or simply "rencontre".

References

[a1] E. Knobloch, "Euler and the history of a problem in probability theory" Ganita–Bharati , 6 (1984) pp. 1–12
How to Cite This Entry:
Montmort matching problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montmort_matching_problem&oldid=39879
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article