# This Ain’t News; Benedict Cumberbatch Is Alan Turing’s 17th Cousin

Well, OK, perhaps it is news that Benedict Cumberbatch is Alan Turing’s 17th cousin because the Mail has decided to give us a story that tells us that he is. Other than that through it really isn’t news at all. What would be much more remarkable is if any two random Englishmen were not 17th cousins, at least.

In fact, we generally assume that pretty much everyone in the UK (except the most recent immigrants) is at least a 12th or 13th cousin. So this finding is in fact really telling us that Cumberbatch and Turing are *less* related than any other two random Englishmen.

He has been praised for his star performance as the code breaker Alan Turing in his latest film role.

But it seems Benedict Cumberbatch’s uncanny resemblance to his subject may be down to more than just good acting…because it turns out they are related.

The actor, 38, is a distant cousin of the celebrated mathematician, who broke the German Enigma code during World War Two.

Both men share a common 15th century ancestor, John Beaufort, the Earl of Somerset, making them cousins 17 times removed on his father’s side, experts from the genealogy website Ancestry said.

The problem with this is that as we go back in time the number of our ancestors increases exponentially. We’ve two parents, four grandparents, 8 great grandparents and so on. By the time we reach 17 generations back we’ve got 131,000 ancestors just in that one generation. And back then the population of the UK was 4 million or so. And yes, that does mean that the odds of any two of us being 17th cousins is higher than 50/50. Much higher: it’s close to a certainty.

The math is here:

To answer the question of “How likely is it that somebody is your 16th cousin” we can just look at how many ancestors you have back there. 16th cousins share with you a couple 17 generations ago. (You can share just one ancestor which makes you a half-cousin.) So your ancestor set from 17 generations ago will be 65,536 different couples. Actually less than that due to duplication, but at this level in a large population the duplication isn’t as big a factor as it becomes later, and if it does it’s because of a closer community which means you are even more related.

So you have 65K couples and so does your potential cousin. The next question is, what is the size of the population in which they lived? Well, back then the whole world had about 600 million people, so that’s an upper bound. So we can ask, if you take two random sets of 65,000 couples from a population of 300M couples, what are the odds that none of them match? With your 65,000 ancestors being just 0.02% of the world’s couples, and your potential cousin’s ancestors also being that set, you would think it likely they don’t match.

Turns out that’s almost nil. Like the famous birthday paradox, where a room of 30 people usually has 2 who share a birthday, the probability there is no intersection in these large groups is quite low. it is 99.9999% likely from these numbers that any given person is at least a 16th cousin. And 97.2% likely that they are a 15th cousin — but only 1.4% likely that they are an 11th cousin. It’s a double exponential explosion. The rough formula used is that the probability of no match will be (1-2^C/P)^(2^C) where C is the cousin number and P is the total source population. To be strict this should be done with factorials but the numbers are large enough that pure exponentials work.

That’s for everyone in the world, not the much smaller number of people who are descended from 15th century England.

There is one thing remarkable about this through. That’s that they can actually trace the connection back to one named individual: that is more remarkable. But the 17th cousin thing? That makes the two less related than the average Englishman is to any other random Englishman.

Posted: 12th, November 2014 | In: Money Comment | TrackBack | Permalink