I want to be King (maths question)

I got to thinking after reading a question about the British Monarchy in GQ…it discusses Queen Elizabeth II’s direct relation to William the Conqueror. I wondered, statistically speaking, how likely I am to be related to William the Conqueror.

The thing is, I have two parents, 4 grand parents, 8 great-grandparents etc, etc…

Assuming approximately 40 generations since 1066 means I have 1,100,000,000,000 relatives that were alive at that time! Obviously this can’t be correct as that is much bigger than the Earth’s population now!

So my question is this, how many great, great,…, great grand-parents am I likely to have that were alive in 1066, what percentage of the Earth’s population (in 1066) is this and hence, how likely is one of them to be William the Conqueror?

A figure that might help…after a quick web search I have found that the population of the earth in 1066 is believed to be between 150 and 300 million.

Cecil has written a column about this:
2, 4, 8, 16 … how can you always have MORE ancestors as you go back in time?