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## Secret Santas I

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #43847 Fun: (2.23) Difficulty: (2.96) Category: Probability Submitted By: javaguru

A group of about twenty friends decide to exchange gifts as secret Santas. Each person writes their name on a piece of paper and puts it in a hat and then each person randomly draws a name from the hat to determine who has them as their secret Santa.

What is the probability that at least one person draws their own name?

1 - 1/e

or approximately 0.63212

where e is the mathematical constant (e ~ 2.71828).

This comes from the number of derangements (permutations in which no element appears in its original position). There are n! ways to draw n names out of the hat. There are [n!/e + .5] derangements of n elements (where [x] is floor x--the integer portion of x). This gives:

[n!/e + .5] / n!

as the probability of nobody drawing their name.

Ignoring the .5 added for rounding (which becomes increasingly insignificant as n increases) this gives

(n!/e) / n!
= n!/e * (1/n!)
= 1/e

Subtract this from 1 to get the probability that somebody draws their own name.

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