How related are Alice and Bob?

The population of Iceland is somewhere around 300,000. It’s not an incredibly big island, and there’s not a huge amount of immigration and emigration – which means, if you’re on the Icelandic dating scene, you have to be fairly careful not to go out with uncomfortably close relatives.

Luckily, there’s an app for that.

However, a chance question from colleague, former Countdown champion, kids’ entertainer and all-round nice guy Barney Maunder-Taylor got me thinking: say Alice Alexsdóttir and Bob Bobsson are getting on nicely at the bar: what are the chances that they’re cousins? Second cousins?

An Icelandic model. No, not like that.

The first thing to do is to look at some data: Wikipedia’s Demographics of Iceland page is quite interesting. A couple of things leap out: firstly, that although the population of Iceland is small, it used to be much smaller (it roughly quadrupled in size during the 20th century); the number of births per year increased rapidly from around 2,000 or 2,500 per year to 4,000+ in the decades either side of World War II; and 93% of the population in 2012 was native to Iceland. I’m going to ignore the other 7%, sorry.

I’m then going to come up with a crude “generational” population model: Alice and Bob are in the same generation, which makes up about a quarter of the population. Their parents are 25 years older, and THEIR parents 25 years older still. And I’m going to make some clearly false assumptions about the nature of Icelandic families and treat the life-cycle of Icelanders as “Age 0: get born; Age 25: marry exactly one person and have (potentially) some kids; Age 50+: get old and eventually die.” This makes the sums easy; it’s not meant to reflect reality, and hopefully doesn’t affect the results too much.

Kissing cousins?

So: Alice and Bob, they have parents. One couple each, as it happens. The population of Iceland 25 years ago was 250,000 or so; the model says a quarter of those people were of reproductive age, which is 62,500, making 31,500 breeding couples – so Alice and Bob have a roughly 1-in-30,000 chance of being siblings. Or they would have, if we hadn’t decided that they’d greet each other with “hey, sis!” rather than “was your daddy a thief?”.

That means, we have two parental units. Let’s dial back to 1963, when the parents were born. There were 186,000 Icelanders, thus about 23,000 parental units (divide by 8 and round). Again, let’s assume Alice’s parents aren’t siblings, and neither are Bob’s. What’s the probability that Alice and Bob share a grandparental couple?

Alice has two of those, since her parents aren’t siblings. Bob also has two. If we call Alice’s grandparental couples couple 1 and couple 2, we can model the problem as a matching-cards puzzle: if Bob picks two cards out of 23,000, how likely is he to pick a 1 or a 2? (Or, potentially, both.) Bob has a 22,998/23,000 chance of missing with the first card and a 22,997/29,999 chance of missing with the second card – making about 1/5,750 that Alice and Bob are cousins.

That’s a small probability, but not all that small – get 90 random Icelanders in a room and the chances are better than 50-50 you have at least pair of cousins in there.

Kissing second or third cousins?

As for second cousins, things get more difficult; you have to take into account the probability of Bob’s parents being cousins (and likewise with Alice’s). Back in 1938, there were 118,000 Icelanders, and just 15,000 or so couples.

If Bob and Alice each had four grandparents (the most likely case, by far), the odds are a little less than 1000-1 against. If one of them has three grandparents and the other four, it’s one in 1,250 – but this only happens once in 3,750 cases. If they each have three (about the same odds as winning the lottery), it’s more like 1/1,700 – making the overall odds of Bob and Alice being second cousins about 1,000-1. In a room of 36 random Icelanders, the chances are pretty good that two of them are second cousins.

For third cousins, it drops to 1/120 or so – and unfortunately, I don’t have any data from before 1900 (Iceland became independent in 1918).

The natural question is: what would similar analysis say about Britain? The answer is, I haven’t done it yet. The probabilities (what with the UK having a lot more people than Iceland) will be much smaller. It’s not super-difficult to write code to work it out though – and I leave that as an exercise for the interested reader.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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