Difference between revisions of "Montmort matching problem"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
''derangement problem'' | ''derangement problem'' | ||
| − | Take two sets, | + | Take two sets, $A$ and $B$, and a [[Bijection|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 paring (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|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 $i=1,\ldots,n$. |
| − | 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". |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Knobloch, "Euler and the history of a problem in probability theory" ''Ganita–Bharati'' , '''6''' (1984) pp. 1–12</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Knobloch, "Euler and the history of a problem in probability theory" ''Ganita–Bharati'' , '''6''' (1984) pp. 1–12</TD></TR></table> | ||
Revision as of 08:09, 23 July 2014
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 paring (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 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=14268
Montmort matching problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montmort_matching_problem&oldid=14268
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article