# Probability of choosing like partners

If there are $x$ (where $x$ is even) people in an office and each person calls out name what is the probability that every worker calls the name of the worker that calls him.

Thanks.

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Obviously it depends on how they choose names to call out. Which names can they call? Just any name? The names of the people in the office? Including their own? Do all the people in the office have different names? According to which distribution is one of the admissible names selected? –  joriki Dec 11 '12 at 10:21
They call randomly –  fosho Dec 11 '12 at 11:47
All of the options I suggested involve calling randomly. Even if you meant "uniformly random", that doesn't answer my remaining questions. –  joriki Dec 11 '12 at 12:34

$(x-1)\times(x-3)\times(x-5)\times...\times3\times1$ distinct pairs of people.

In each one of these pair instances, each specific pair call one another's name by the probability of $\frac {1}{(x-1)^x}$ assuming no one calls his own name.

$\frac{(x-1)\times(x-3)\times(x-5)\times...\times3\times1}{(x-1)^x}$

$x$ people ($x$ is even) forming $(x-1)\times(x-3)\times(x-5)\times...\times3\times1$ distinct pairs: In these pairs, any one of the $x$ people could have paired with $(x-1)$ others. An arbitrary other among those remaining can appear in this set of pairs with $(x-3)$ of the remaining persons and so forth.
Assuming each can call one another's name with equal probability, a person can call out anyone of the $(x-1)$ names except his own. The probability of each person calling the name of the person he paired up with is $1/(x-1)$. Each one of the $x$ people calling out the "right" names in a set of pairs is $1/(x-1)^x$. Multiply this by the number of such pair instances.