Revisiting the birthday problem

03/20/2012

If you've ever taken a statistics course, you've probably heard of the birthday problem. If you haven't, the birthday problem asks what the chances are that two people in a room full of, say, n people have the same birthday. It's a fun problem to solve in a class of 23 or more people because it turns out that there's a greater than 50% chance that two out of 23 individuals were born on the same day. More importantly, it turns out that nearly 100% of probability theory newbs think the chances are much lower.

Also, I pulled one of those statistics out my butt. Guess which one, then realize it might not be far off the mark and forgive me.

Today, I was reminded of the birthday problem while I was reconciling church birth records with a genealogy for this village, which I've been improving since 2010 by combining key informant interviews with archival research. One of the last records I checked was a woman whose first initial is J., and who was born on my birthday in 1927, 57 years before I was born.

"Wow," I said to myself, "I found someone with my birthday, what are the o..."

At that moment, the Debbie Downer of statistical reasoning blew her sad trumpet to remind me of the birthday problem, which I remember from one of my game theory courses in which we reviewed probability theory before getting into the Bayesian equilibrium concept. No, Brash, sighed statistical reasoning, it's not that special.

I felt like most women feel when I them that the whole menstrual synchrony thing is (probably) complete bullshit, as Cecil Adams summarizes with his usual aplomb.

I slapped myself out of my statistical-reality-induced funk and thought, "Actually, it's kind of cool that I'm almost guaranteed to share a birthday with one of these people." Then I hopped over to Wikipedia to review how to calculate the probability to avoid thinking hard at midnight after 12 hours of work. Finally, I calculated the vanishingly small probability that I don't share a birthday with any of the 1,326 records in my genealogical database.

I'll skip the details and tell you that the probability is zero. Here is a good example of how scaling up a problem can help you understand it. We can reduce the birthday problem to the absurd and envision the probability that I don't share a birthday with at least one other person on Earth, which currently nourishes about 7 billion sacks of living, breathing, human meat. But you'd think I was an idiot if I thought about calculating the probability in that case. Of course someone shares a birthday with me. Hell, on my birthday this year, about four people will be born every second.

Thank about that. The second you were born, three other beings were likely screaming their way into this confusing, beautiful world. Each birthday you have you share with enough people to enlist in almost 4,000 Roman legions.

I only wish that the scaling up problem worked in my personal life. Alas, I have an n of only three. The simultaneous trials and triumphs of parenthood, grad school, and living apart from my family must remain seeming paradoxes until I get the time to sort them out the old fashioned way.

What?

My n of 3 once told the true (and sometimes fictional) story of an anthropologist and Fulbrighter who (kind of) became what his doctoral dissertation is about: a migrant trying to maintain his ties with family back home. It turned into a dissertation blog. Then it turned into a science blog focusing on human evolutionary ecology, especially the evolution of cooperation, aggression, warfare, and social inequality.

Who?

Brash Equilibrium is an evolutionary anthropologist, quantitative researcher, and Fulbrighter whose real name is Ben Hanowell. He lives with his daughter in Ballard, Seattle, Washington, USA. He's also building a company called SoundCheks, which he hopes will change the face of political discourse.