Proof of royal blood

I have royal blood. I'm a direct descendant of one of the great kings in the history of the world. I am sure of it. Pretty cool eh? 

Which one, you ask? Well, ahem, I just mean I'm mathematically almost sure I am a descendant of a great king. And so are you.  

Here's the proof. First, like everyone, (except clones if there are any)  I have 2 biological parents, 4 grandparents, ...,  2^n direct ancestors n generations back. This is not an estimate, it is a precise fact. Second, I am not a descendent of extra-terrestrials, so I must have human ancestors on earth who were around at all points in human history. Now  let's take say n=30. That would be about 1000 years ago.  That means I have 2^30 that is about 1 billion ancestors that existed then. Woah. But the world population was only a few hundred million. So how is that possible? How can you fill a billion positions on my family tree, which, remember *must* exist, when there are only a few hundred million humans in existence. Obviously it's because the same person must fill multiple slots. If there was 100M people  let's say, then each one on average must appear 10 times. Of course, some get more than average and some get less. It's like a lottery. And the odds of each person on earth getting many slots in my tree are directly related to how many children, grandchildren etc they had. Obviously most kings have more descendants than the average person by far. Not only do they have more kids, their kids are more likely to survive and so on. Therefore in that lottery if the average human appears ten times then the average king who existed 1000 years ago must appear way more than 10 times. Therefore I have lots of kings at n=30.  Now if we keep going, as n gets larger, each king gets exponentially growing number of lottery tickets to win places in the tree. If you add it all up, the probability that I have at least one great king approaches 100%. QED.

P.S. I first read a version of this argument in a magazine article years ago, I think it was Harper's or something, will try to find it and give credit.

P.P.S. The picture is of King David (I love how it looks like "Dawit" but I think it actually says "Davidus"), from the Nine Worthies in the Palazzo Trinci, Italy, circa 1410.