How Many Kids Did Someone Have If I Know How Many Grandkids They Have?

I recently overheard a coworker mention that they were the youngest of 20 grandchildren. I’d heard numerous similar huge grandkid counts over the years, and for some reason, I always try to figure out how that final number was arrived at. Did the grandparents have two kids who each had ten kids of their own? Ten kids with two kids each? Four with five? I would only know if I asked the simplest possible follow-up question, but that isn’t too much help when you’re eavesdropping. So instead, I’m going to use my time to figure out the answer probabilistically.

How do I do that?

Helpfully, it is pretty easy to find the average number of children born per woman.

Unhelpfully, it is a lot more difficult to find the distribution of children. If there are two and a half children per woman, that could mean half had two and half had three, it could mean an equal number had one, two, three, or four, it could mean seventy-five percent of women have no children and twenty-five percent pop out ten kids. We need more information.

Helpfully, I was able to track down some Pew Research data on the spread between different family sizes.

Unhelpfully, it (and every other source I could find) grouped “four or more” together, meaning that I would need to estimate just how many had four, how many had five, and so on.

We now venture into the unknown.

Assumptions

• Everyone involved is done having kids. I tried to make this work without this assumption, but I ended up deep in the weeds over current ages, gaps between kids, etc. Like, if someone had no grandkids, it meant they were probably too young to have any kids. If they had one, they probably just became a grandparent. I tried, guys.
• The grandmother was born between 1932 and 1936 (Pew Research data available) and for average births per woman, we’ll use 1965.
• Her children were born between 1960 and 1964 (Pew Research data will use average between 1994 and 2014 40-44-year-olds) and for average births per woman, we’ll use 1993.
• Number of kids above four decreases at a steady exponential rate.
• Maximum number of kids is ten. I needed an upper bound and arbitrarily set it here.
• Number of kids in each family unit is independent of each other family unit. Your parents or siblings having many kids does not influence how many you have.
• Everyone involved survives through their fertile years.

Completing the Family Size Distribution

First up, we need to find the childless population for each generation. Per Pew Research, we will be taking 10% for the grandparents and 16% for the parents.

Using the earlier chart of the percentage of mothers who have different numbers of kids and this one that lets us know how many become mothers, we have an idea of how many one, two, and three child families there are. To figure out four through ten, we had to set the total percentages equal to 100% and we had to make sure the weighted average of each family size equaled the observed average, 2.93 in 1965 and 2.02 in 1993. Miraculously, this occurred when the family size was set to decrease in probability at 55% per additional kid for both the grandparents and the parents.

How Many Grandkids for Each Number of Kids?

Ah yes, the opposite of our question. Once we solve this, we can use the probability of the grandparents having each number of kids to answer our original question.

Right off the bat, no kids means no grandkids and one kid means that the probability is equivalent to that of the parent having kids probability. Where things get interesting is if the grandparents have two kids. Now to figure out the odds of them having three grandkids, we need to find the probability of:

• Child A having three and Child B having zero
• Child A having two and Child B having one
• Child A having one and Child B having two
• Child A having zero and Child B having three

Then we add them all together. This is what the odds are for each number of grandkids given two kids:

To calculate the odds for each number of grandkids if you have three kids, instead of starting at square one and multiplying each of the three kids independently, we can look at the sum of the first two and the third. Then for four, we can look at the sum of the first three and the fourth. This cuts down our permutations from 11 x 11 x 11 x 11 to just 11 x 31. Without this trick, solving the number of grandkids if you had ten kids would take over 25 billion permutations. With it, just 1,001.

How Many Kids for Each Number of Grandkids?

Now that we have our distribution for each number of grandkids for each number of kids, and the probabilities for having each number of kids, we can finally tackle the original problem. Say we wanted to know how many kids someone had if they had five grandkids. Having one kid that had five kids or having five grandkids split amongst four or more kids are both relatively unlikely.

If you had two kids, the likelihood they would tally five grandkids is 17.091%. If you had three kids, the odds of five grandkids is 16.663%. You may want to jump to two kids as the answer, but we have to remember, we have to take into account the probability of having two or three kids in the first place. For the grandparent generation, the odds of two kids is 21.6% whereas the odds of three kids is 22.5%. This means the odds of having two kids who had five grandkids is 3.692% and the odds of having three kids who had five grandkids is 3.749%.

Using this logic, we can determine the most likely number of kids for each number of grandkids. Without any further ado, here is that table of how many kids you should guess for each number of grandkids:

It appears that for higher numbers of grandkids, you’d do best to assume each kid averaged a little under three children of their own. I’ll take it.

And now, I’ve got to go ask that coworker what it was like for his grandparents to raise seven kids.