# Human nature, how we teach math, and the birthday problem

I’ve spent a few weeks wondering what I can write about for my first post here. I’m a first-year PhD student with an endless supply of questions but without much wisdom or insight to share yet about my short graduate life. As a recent college graduate, however, I have spent years thinking about how my friends and peers perceive my mathematical interests and my career choices. And while I’m still learning about graduate life, I have a wealth of opinions about how we approach communicating math and statistics to students, clients, and colleagues interested primarily in other areas.

In college, I asked many other students to clarify why they don’t like math. I got two answers over and over again- it’s boring, and it’s too hard. The chain rule and the shell method of integration do not strike them as relevant to their future careers or to their broader understanding of the world around them. Moreover, they have been told from a young age that math is too hard, so why bother trying?

As mathematicians, we appreciate beautiful math for its own sake and do not question that a theorem’s truth accords it value. If we incorrectly assume that our friends and our students automatically share this appreciation for mathematical truth and elegance, however, we miss the opportunity to reintroduce them to math in a way that challenges their views on it.

What got you interested in math in the first place? For me, an important part was questions that have surprising answers, questions that make me think differently and re-examine the world around me. A simple but incredible example of such a question is the famous birthday problem. Given 20 people in a room, what is the probability that at least two have the same birthday? Intuitively, most people would think the probability is fairly low. Ignoring leap years for simplicity, there are 365 days in a year, so 20 people seems like a very small number comparatively; it should be pretty unlikely for 2 to have the same birthday.

As you probably know, an easy way to actually calculate the probability is to compute the probability that no two people have the same birthday, and subtract it from one. Take the first person in the room- let’s say it’s me. My birthday is August 18th, so we can rule out August 18th for everybody else. Then consider the second person in the room; we must have different birthdays, so he can have a birthday on any of the remaining 364 days, which happens with probability 364/365. Say his birthday is September 26th. Now take the third person; she can have a birthday on any of the remaining 363 days, which occurs with probability 363/365, and so it continues. We multiply each of these probabilities to get our answer for the probability that no two students have the same birthday: ∏ (365-i)/365, from i=1 to i=19. Call this probability A; then,  the probability we want is then 1-A. I (okay, Wolfram Alpha) computed this as about 0.411438.

Who would have guessed? With only 20 people there’s a 2/5 chance that a pair of them have the same birthday out of 365 possible birthdays! This seemingly paradoxical answer is mathematically satisfying, but also reveals some interesting ideas about how we think (Stamp, Mark. Information Security: Principles and Practice. Jon Wiley & Sons, Inc. Hoboken, New Jersey, 2011). Why are our first guesses so far from the truth?

My first instinct, a common one, is to think about the probability that somebody else has the same birthday as I do. That’s 1/365 * 19, about a 5% chance. The less obvious but far more important question is: do any of the people in the room share a birthday with each other, that may not be August 18th? When we consider that, we open not just 19 comparisons but 20 choose 2, or 190 comparisons. In that light, it seems far more likely that we could get a match; now 41% sounds quite reasonable.

The birthday problem is a fairly straightforward math exercise. However, the seemingly paradoxical answer to it highlights our nature to think selfishly, to insert ourselves into every comparison even though we should mostly be considering pairs of other people that do not contain ourselves. A counting problem concerning days of the year may be difficult, but it is far more approachable than taking anti-derivatives of complicated functions. And considering the selfish nature of human beings, and how it manifests itself in our attempts at problem solving, can hardly be called boring or worthless. If we can present math and statistics in a framework like this, we can engage far more students. Math is surprising and amazing sometimes, and while elegant proofs have merit, so do tricky and surprising problems. Both perspectives on math were central for developing my excitement about the field, and have motivated me to pursue a graduate degree in statistics. So here I am, and for everybody with wisdom to share about qualifying exams, choosing an advisor, and getting that NSF GRFP, I’ll be reading.