The Pregnant Mathematician Continues Teaching, Probably

Recent Events
During the month of June, I’m teaching our “Introductory Calculus (Math 120)” course. We meet five days a week for 145 minutes, with a total of twenty class sessions running June 5th through July 2nd. Our final exam will be held on Wednesday, July 3rd. A few days ago, my very compassionate students expressed interest in knowing the probability their final exam would be cancelled due to an unexpectedly early birth of my daughter. I’m sure their inquiry was based solely on concerns for our health and not at all from them hoping to escape the intellectual challenge of a cumulative final exam!

Also recently, my husband and I went on an “Expectant Parents Tour” of the hospital where I will be delivering. Our son was born in November 2010, but at a different hospital; this time, I will be delivering at a hospital that was still under construction then! Since they have recently opened their doors, their NICU [Neonatal Intensive Care Unit] is still a “Level 1” facility. This means that they are certified to care for healthy infants born at 36 weeks or later (measured from LMP) or 34 weeks gestational age (i.e., since conception). Infants who are born with complications, or who are born before 36 weeks, are usually transferred to another local hospital. It turns out that reaching “Week 36” of my pregnancy and having lots of calculus final exams to grade will coincide perfectly.

Some Probability
While on our tour, I started wondering about the chance that I would go into labor early enough that it would affect my summer class. Not only would this be an unfortunate inconvenience for my students, it would also mean that I would probably have to deliver at a different hospital since I wouldn’t be at 36 weeks yet. What’s the chance this happens?

First, I should point out that I’ve had a relatively uneventful pregnancy — thankfully, both my unborn daughter and I have been in excellent health (if not a little grumpy from having to share the same circulatory system and oxygen supply). Second, my son was born within a few days of his due date, a little on the early side. Third, I’m not carrying multiples, nor am I expecting any major complications as my due date approaches. So we will just assume that this is an average pregnancy as far as the medical issues are concerned.

One of the things I talked about in my original “Pregnant Mathematician” post was how due dates are calculated using Naegele’s rule. Also, there was a rather large (n=427,582) study done in Norway [See Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.] that found the mean gestational length for singleton pregnancies was 281 days, with a standard deviation of 13 days.

Let’s assume a mean of 281 days and a standard deviation of 13 days. What’s the chance a woman goes into labor 251 days or earlier (corresponding to 36 weeks)? Notice that 251 is about 281+(-2.31)*13, so giving birth prior to 36 weeks means you’re about 2.31 standard deviations away from the mean. By the Empirical Rule, I know this would be quite rare: There’s less than a 2.5% chance!

We can use Wolfram|Alpha to compute the exact probability. Our input is the command “CDF[NormalDistribution[mean, stdev], X]”; in this case, we are taking mean=281, stdev=13, and X=251. Wolfram|Alpha returns a result of 0.0105081, meaning there is about a 1.1% probability that I will give birth early enough to impact my current students.

It’s Gonna Be How Hot?!!

According to tomorrow’s weather forecast, the heat index here will reach 110 degrees tomorrow. This is miserably hot for everyone, but especially if you’re pregnant. Although my actual due “date” is in the first few days of August, I’m sincerely hoping that the birth occurs sometime in July. August heat in South Carolina is no joke! Wolfram|Alpha comforted me with its computation that there’s about a 41% chance I will give birth before I have the opportunity to be pregnant in August.

I’m also going to assume my chances of a July delivery are even higher than this, since human gestation doesn’t exactly follow a normal distribution. While a measurable percentage of moms give birth two or more weeks early, nearly none give birth two or more weeks late. By that point, OBs usually induce labor because of declining amounts of amniotic fluid and concerns for the health of the newborn. I’m going to take this and a “fingers crossed” approach and assume that a July birthday is at least 50% possible.

[There’s a name for such a distribution; I thought it was a truncated normal distribution, but that doesn’t seem to be quite right. The statistician who told me the term isn’t in his office at present! Anyone know what it’s called?]

Postscript: Dear Students,
Regardless of when I deliver, I can assure you that your calculus final examination will occur as scheduled. I know lots and lots and lots and lots of people who really enjoy torturing unsuspecting college students with tough calculus exams, and it would be easy for me to cajole one of those people into proctoring your test! So, don’t fear: You will certainly have the opportunity to demonstrate all the calculus you have learned this summer & feel proud of your scholastic achievement upon completing our course — including its final exam. 🙂

Probability and Weather

Warning: I know very little about probability. I know even less about weather phenomena! The post below describes something I was thinking about today, because I find it interesting and I’m procrastinating when I should be grading a giant pile of calculus exams instead!

The following image appeared on my Twitter feed, courtesy of @LCWxDave:

This made me wonder, “What is the probability that at least 21 of the last 27 day’s highs have been at or below average?

First, let’s make two (probably bad) assumptions: (1) High temperatures are normally distributed, and (2) the events “Today’s high temp” and “tomorrow’s high temp” are independent. If this is the case, then the probability requested above is

(27!/(21!6!)+27!/(22!5!)+27!/(23!4!)+27!/(24!3!)+27!/(25!2!)+27!/(26!1!)+27!/27!)*(0.5^(27)) = 0.0029623

or about three-tenths of one percent. This struck me as being exceedingly rare! Then Dan Jarratt pointed out,

It puzzled me that my computed probability was 0.3% but the actual collected data suggest happened 8% of the time! What’s going on?

First, I still have no idea about my assumption about temperatures and normal distributions. Second, I really ought to do a more careful calculation and not treat each day’s high temperature as independent from the next day’s high. Surely, if it’s very cold on Wednesday, it is probably pretty likely it’s going to be cold on Thursday, too.

So instead of treating the 27 days as 27 different events, let’s consider them as 13 two-day events. Out of these 13 two-day events, about 10 of the two-day events have been colder than average. New question: “What is the probability that at least 10 of the last 13 two-day’s highs have been at or below average?

In this case, the computation yields

(13!/(10!3!)+13!/(11!2!)+13!/(12!1!)+13!/(13!))*(0.5)^(13) = 0.046142578125

So my computed probability is 4.6% with the data suggesting 8%. The comparison between these two seems far more reasonable. What this tells me is that today’s weather seems quite dependent on yesterday’s weather, which isn’t surprising. After discussing this with my colleague Garrett Mitchener, he pointed out that a great way to predict tomorrow’s weather is to say that it will be exactly like today. Hopefully, we are better mathematicians than weather predictors.