A few months ago, I read an article about a family with 14 children, all boys. That is a lot of boys! My first thought was what their grocery bills must be. (Whether they’re all boys or not, a family with 14 kids has multiple teenagers for decades. I remember fondly the damage I could do at my favorite all-you-can-eat pizza buffet when I was 15.) My second thought was that having 14 boys in a row seemed statistically improbable. But how improbable was it? I didn’t know whether or not I should be surprised that there is a 14-boy household in the U.S.

To first approximation, a child has a 50 percent chance of being assigned male at birth and a 50 percent chance of being assigned female at birth. (I will call those categories boys and girls, respectively, though both sex and gender are more complicated than that.) So having a boy or a girl is about like flipping a coin. It’s pretty unusual to flip 14 heads in a row. Then again, any particular sequences of 14 heads and tails, or 14 boys and girls, is equally probable. If I had read about a family whose kids were girl-girl-boy-girl-boy-boy-boy-boy-girl-girl-boy-girl-boy-girl, I probably wouldn’t have marveled at what an improbable birth order it was. (Holy semantic satiation, Batman! After typing it so many times, I just looked up the word “girl” to confirm it is actually a word that exists and is spelled that way.)

Refining the probability a little bit, there are about 105 boys born for every 100 girls, so about 51.2 percent of babies are boys. The probability, then, of 14-boy families among all 14-child families, is (.512^{14}), which is about 0.0085 percent. Rounding that to the nearest order of magnitude, we would expect about 1 in 10,000 14-child families to have only boys.

After working out that probability, I had a new question: how many 14-child families are there in this? If there are more than 10,000, it seems likely that there are some 14-boy families out there. Even if there are only a few thousand 14-child households, it wouldn’t be too surprising to have a 14-boy family.

At this point I turned to the Census Bureau. According to the report I used, there are approximately 35 million households with children under 18 in the U.S., and about 2 million of those have at least 4 children. Unfortunately, the tables I found do not give a more detailed breakdown of family sizes. But there are a few ways we can at least try to get a better sense of things. First, if those 2 million 4+-child households all had 14 children, we would expect to have a couple hundred 14-boy families in the U.S. That is obviously a vast overestimate. There are many more 4-, 5-, and 6-child households than 14-child households. What proportion of 4+-child households have 14 children?

The only more granular information I could find was from a 2011 New York Times interactive about how common various household compositions are. As you add more children to a family, you can see the number of such families dwindle. The interactive will not let you add more than 10 children, either under or over 18, to the family; there are around 1,100 10-child families in the U.S. (The precise number varies depending on how many of the children are over 18.) I feel somewhat confident based on that data that there are fewer than 1,000 14-child families in the U.S., and that makes me think the Schwandt family is indeed statistically improbable.

There is one assumption I haven’t really explored in the above estimates, which is the assumption that in a given family, the sexes of two children are independent events. In other words, I assumed that each child had a 51.2 percent chance of being a boy. But perhaps some families are more likely to have boys and some are more likely to have girls. Some couples use something called the Shettles method to try to increase their chances of getting one sex or the other when they conceive, possibly changing the probability of getting a boy. And there are some studies that suggest there may be a small bias towards one sex or the other within some families. Mikhail Monakhov’s paper on the topic reports that there are more all-boy and all-girl large families than expected. The largest family size they reported on was 10 children, and there were about 1.6 times as many 10-boy families as expected.

After exploring several different aspects of the problem, I have to conclude that a 14-boy family is on the unlikely side, so I stand by my initial surprise. I think it’s safe to assume the Schwandt family is the only 14-boy family in the U.S. Now that I’ve satisfied my curiosity on the matter, I can go back to worrying about the bottomless pits that are teen stomachs.