Difference between revisions of "Montmort matching problem"
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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 permutations $\pi$ such that $\pi(i)\neq i$ for all $i=1,\ldots,n$. | + | 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 |
Montmort matching problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montmort_matching_problem&oldid=39757