[dba-SQLServer] sql server 2005 database mirroring - how many copies of sql server?

Hello:

I need someone to clarify database mirroring for me.  How many copies of sql 
server do I need to configure a database mirroring session involving three 
separate servers (that includes a witness server)?

The scenario is this: in the quorum, there are three servers: the principal 
server, the mirror server and the witness server.  the principal server has 
the original db; the mirror server has a copy of the db on principal server; 
the witness server watches both the principal server and the mirror server.  
If the principal server goes down, the witness automatically tells the 
mirror server to take over as principal server, a process that takes 3 
seconds.

To establish the mirror session for all three servers, according MS 
eLearning course 2938, I perform the following:

For the principal server's participation in the quorum:
1) from the principal server, I run the alter database statement on the 
original db

For the mirror server's participation in the quorum:
2) from the mirror server, I run the alter database statement on the mirror 
db

For the witness server's participation in the quorum:
3) from the PRINCIPAL server, I run the alter database statement on the 
original db

Given step 3), I can't tell if I need 2 or 3 copies of sql server (ie. do I 
need a copy of sql server on the witness server?)  Am I missing something 
here?

Thanks in advance,
Billy