Associated Topics || Dr. Math Home || Search Dr. Math

### Probability of Getting your Hat Back

```
Date: Mon, 09 Oct 1995 16:36:05 -0400
From: chris mcmaken
Subject: Hat probability

There are 5 people with 5 hats. The hats are put into a box.
What is the probability that each person will get his or her hat?
```

```
Date: 5/30/96 at 14:32:10
From: Doctor Charles
Subject: Re: Hat probability

The probability that the first person will get the right hat is
1/5. Now there are four people and four hats so the probability
that the second person gets their own hat is 1/4. Similarly for
the third and fourth people. Of course, if the first four people
all get the correct hat then the last person must get the right
hat too as it is the only one left.

The total probability of everyone getting his own hat is found by
multiplying all these together.

That is (1/5)*(1/4)*(1/3)*(1/2)*1 = 1/(5*4*3*2*1) = 1/120.

Incidentally, the function that gives n*(n-1)*(n-2)* ... *3*2*1 is
called the factorial function and is written n! . In general n! is
the number of ways of placing n objects in a row. So if we had n
people with n hats then the probability would be 1/(n!).

-Doctor Charles,  The Math Forum

```
Associated Topics:
High School Probability
Middle School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/