## Genealogy Problem

Problem of the Day: Just to keep our minds nimble through the summer I thought I’d post a little probability puzzle. My dad has recently been studying genealogy. He remarked to me yesterday that in one line of our family tree the same last name persisted for 500 years. I was amazed at first, but then began to wonder what the expected number of generations a last name sticks around for. So here’s the setup/assumptions of this toy problem.

- Each couple has C children that marry and reproduce.
- There’s equal probabilities of those children being male or female.
- Females don’t carry on the last name (there’s no incest, females marry males with different last names, and wives always take husbands’ last names).
- We start with a married male with the last name EconLove.

Part 1: Work out the probability that the name EconLove will survive for at least Y additional generations. (Hint, I didn’t have a reduced form equation, but instead a recursive one.)

Part 2: For C=2, what is cut-off number of generations where the probability of EconLove surviving that number of generations drops below 1/2? For what values of C will the probability of EconLove lasting Y generations never drop below 1/2 for any Y? (Hint, there’s an intuitive reason for this last one).

I’ll wait a couple days to see if people have questions, and then I’ll post my answers.

Curb the enthusiasm guys.

Part 1: We start with a subproblem. First imagine an upside-down tree of G generations descending from a root person, with each person in the tree being either male or female and having C children. The last name “survives” if there is a chain from top to bottom that’s all male. It does not survive if all chains contain at least one female. We will call this latter probability f(G), with f(G-1) being the probability that there is a female in all chains that descend from a child of the original root person. Now we look at it recursively. There’s two ways for the name not to survive, either the root person is a female (.5 chance) or the root is a man and there is a female in all chains that descend from everyone of the man’s C children (f(G-1)^C chance). We can then write f(G)=.5+.5*f(G-1)^C with f(1)=.5.

Then if we start with Mr. EconLove, the probability that his names lasts Y generations is 1-f(Y)^D.

Part 2: Plugging all that into the computer, the probability of the name continuing-on drops below 50% between the 3rd and 4th generation. If C>2 then the probability of the name living on always stays above 50%. This is understandable because with this condition the population is always growing.

bequwAugust 21, 2009 at 10:12 pm

Sorry Brian! I’ve been on the train the past week with no internet. I’ll be more active now 🙂

jkAugust 24, 2009 at 9:18 am

Hey Brian,

Isn’t the key part of this problem that the number of children people have is variable? The reason names are likely to die out is that some people have no children at all. I don’t think this makes the problem that much more complicated – number of children can be drawn from an poisson or exponential distribution with a mean of 2. Having said that, I don’t have an answer.

TomSeptember 5, 2009 at 12:05 pm