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