I'll be honest, I'm a lazy b*****d, and the amount of work involved in applying to Oxford or Cambridge put me off.
However, you cannot deny the following.
* The application process is intense.
* It is also quite different to applying to any other UK university.
* You are more likely to have teachers who graduated Oxbridge in a private school than a state school.
* Those who have graduated Oxbridge are more likely to understand the application process, and will be more helpful to those applying.
This does, intrinsically, give those who had a private education a greater chance of gaining entry to Oxbridge.