Monday, May 30, 2005

Statistics of Ancestry

So, just in case J. Brad DeLong's blog and entries ever go poof, I want to post this lovely story he wrote here.

December 08, 2003

One Hundred Interesting Mathematical Problems, Puzzles, Diversions, and Amusements: Number 21: Ancestors



"Were my ancestors... famous back in the Middle Ages?"

"Your ancestors? Which ancestors? You have a lot of them."

"Well, most of them."

"There is a tradition that your Ridgeway ancestors--part of the English wave of conquest of Ireland under the Angevin kings--were descended from an illegitimate daughter of William the Conquerer, and that they were also descended from Earl Leofric and his wife Godiva, of the well-known story about Coventry. This was possibly the first supply-side tax policy in history: she asked her husband to lower taxes, and he replied 'Sure, I'll lower taxes when you ride naked through the streets on a white horse'."

"Nobody knows that story, Dad."


"I asked people in my class. They all agreed that Lady Godiva was the inventor of a kind of chocolate."


"But what was my typical ancestor like?"

"Well, first let's calculate how many ancestors you had. You were born in the 1990s. Let's suppose that there is a generation born every twenty-five years. That means that a century ago you had 2 x 2 x 2 x 2 ancestors born in the 1890s. How many is that?"


"In exponential notation: you're studying exponents in math this week."


"And a century before that? Another four generations, with your number of ancestors doubling as we go back each additional generation?"

"You'd multiply by two four more times."

"Which would give you?"

"28, two multiplied by itself eight times."

"That's 1790. How about 1690?"


"And what is 212, approximately?"


"Well, 212 is 210 times 22, and 210 is about 1,000--1024 exactly, but we'll say 1000."

"1000 times 4 equals 4000."

"Yep. About 4000 ancestors born in 1690. Now let's take it back to 1490--eight more generations. 212 times 2 eight more times is?"


"And back to 1240--another 2 1/2 centuries, another ten generations?"


"And that is? Remember, 230 is 210 x 210 x 210."

"1000 x 1000 x 1000... is a billion."

"Excellent! Now consider that there were only 400 million people alive on the earth in 1240. What does that mean?"


"Well, it means that you must be descended from a lot of people by a lot of different lines of inheritance--a lot of distant cousins marrying each other. I mean, there are a billion ancestral slots that have to be filled, and only 400 million people back in 1240 to fill them. So a lot of people have to be filling multiple slots on your ancestral family tree."

"So does that mean I'm descended from everybody alive in the world in 1240? That they are all my ancestors?"

"Not quite. There were a number of people alive in 1240 who have no descendants living today. And mixing between European populations and the Amerind, sub-Saharan African, Indian, southeast Asian, Chinese, and Japanese populations over the past millennium has been very slow. But if they (a) have any living descendants at all, (b) were alive in 1240, and (c) lived in Europe, on the south or eastern shore of the Mediterranean, or in Mongolia, odds are that you're probably descended from them."


"Your mother's maternal grandfather had Tartar eyes. Mongolia. Chingis Khan. The Mongol Empire."

"So when I read about the history of Medieval Europe, I should think that everyone's my ancestor?"

"Yep. The Holy Roman Emperor Friedrich I Barbarossa of the House of Hohenstaufen, and also the peasant Ludovico il Pazzo from Caserta outside Napoli. Pope Innocent III, or at least one of his close relatives. William the Conqueror and a number of his knights, but also Aethelraed Illraed and Harald Hardrada. Lady Godiva, but also Brunhilde the washerwoman, and Thraxa the house slave back after the Roman and before the Saxon conquest of Britain. Rich and poor, noble and serf, pretty and ugly, smart and dumb--all of them, at least all of them who have any living descendants at all today."

"Kind of stupid that they spent so much time fighting each other, isn't it?"