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## getting our own mugs

### Stuart Altmann on mon 13 oct 97

"What would be the odds of everyone getting their own mug back?" --Martin

Suppose that n people participate in the Clayart mug exchange, and that we
decide who gets which cup by each drawing at random a number that
represents the cup that we get. The first person who draws a number has n
possible outcomes. For each of these, the second person has n-1
possibilities (because one number has been removed). The third person has
n-2 possibilities, and so forth, down to the last person, who has but one
possible number to draw. Altogether, there are n(n-1)(n-2)...1 different
ways that the n numbers could be drawn. Of these, only one would result in
everyone getting back their own mug. So the chance of that happening is 1
out of n(n-1)(n-2)...1.

To illustrate, suppose only 5 people participate in the mug exchance. Then
the probability that everyone will get back their own mug is 1 out of 120,
that is, it would on average happen only 0.8 percent of the time. If 10
people participate, it would happen only 0.00000027 percent of the time. For
any number of participants much beyond 10, the probability quickly goes from
being tiny to being miniscule. Bottom Line: it won't happen. Want to bet
on it?

Stuart Altmann