# what the birthday problem can teach us about when to have a baby

Carl Bialik at FiveThirtyEight recently posted a chart that summarizes when babies are born, which reflects attitudes and practices about scheduling inductions and C-sections. The chart shows that people are superstitious and have fewer babies on the 13th of each month, particularly on a Friday the 13th. I am not superstitious. My birthday is also on a 13th, although I was born on a Thursday. My mom had no choice in the matter because she went into labor naturally. But I’m confident she would not have shied away from the 13th.

This chart has implications for The Birthday Problem:

Given a group of n people, what is the probability that someone shares a birthday?

The canonical problem assumes that people are equally likely to be born on any day. If some days are more likely than others, then a match is even more likely. I simulated the birthday problem with a “mating season” here.

The data can be used to identify when hospitals have room shortages and when to have extra staff on hand. When I saw Bialik’s paper, I couldn’t help but think that the 13th of the month might be the ideal time to have a baby. Or the 15th. Or January 3. Or July 4.

We live in a world of limited resources, where we run some risk of not having enough of what we want. The above chart suggests that risk is not evenly distributed to all births. Having a baby on the 13th means that there is always a delivery room ready. If the hospital only has one bath for water births, you may have less competition for it. Or not. Having a baby on the 15th means that women who had babies on the 11th and 12th were discharged, leaving plenty of rooms in the maternity ward for mom and baby to recover.

When I had my third daughter, the hospital was out of rooms in the maternity ward. Only one woman was discharged, which meant that only one mother could be admitted. I was selected for the room because my daughter was born at home in a precipitous labor and accidental home birth. The staff felt that since I did not use a delivery room, I should get the room in the maternity ward. I was grateful.

This is not uncommon. You will always get a room, but you might have to wait in an uncomfortable place, such as a delivery room or elsewhere. I stayed by myself in the delivery room for hours after my second daughter was born instead of recovering in the maternity ward. It was stressful because I could not see my daughter for a long stretch of time, because security dictated that babies in the maternity ward could not leave. I was eventually taken to the maternity ward, where I discovered that my daughter was doing just fine and sleeping peacefully.

Mothers will always get a room somewhere eventually. Most women stay one to four nights in the hospital, depending on the details of the birth and their preferences about staying at the hospital if things are going fine. A hospital can be less busy for one to four days after a lull in births on the 13th of the month. I dealt with a hospital shortage after 2 of my 3 daughters’ births. None of their births were ideally timed according to my reasoning above. And none of their births were poorly timed on a busy day (see the heat map below)

So far I’ve talked about room shortage. Rooms are one type of limited resource. Staff are another. Staffing can be changed in response to trends in births, but hospital rooms cannot since there are only so many rooms in the maternity ward. Still, extra staff can make things happen to make the hospital run smoother when it’s busy. Discharges can be processed faster, for example, which can address a shortage of rooms.

I love talking about my daughters and analytics 🙂 Anyway, here is a heat map of birthdays.

A heat map of likely birth dates