Day 769: Happy Birthday To Possibly Both Of You

originally published February 7, 2014

Every so often I like to write an article about math. 

This is not because I like math (I don’t) or because I unearth a specific gem of unique adrenaline when I reach the conclusion of a math problem (I experience no such thing). I write about math simply to remind myself that it’s there, and to give myself the daunting challenge of making some portion of it interesting. You thought writing a thousand words in Shakespearean sonnets was tricky? That was a seat at the sundae bar compared to this. 

I took my last math class in high school. And by ‘took’, I mean I sampled bits of it, but left most of it on the plate. Once the curriculum started to include concepts like quadratic equations and logarithms, I allowed my attention to meander to more earth-bound notions, like old soul records, pretty girls and caramel sauce (sometimes all three at once – I was a creative kid). I simply don’t have a brain that is hard-wired for that brand of thinking. 

But I’ve danced down this hallway before, poking my head through the door of Monty Hall’s Let’s Make A Deal 3-door conundrum. And even for us non-mathies, sometimes numbers can take our brains by the squishy parts and lead them on a fun little trip. 

Where math tends to poop out its most delectable little gems is in the realm of probability. How the oddsmakers in Vegas determine a 2.5-point spread for a game instead of a 3-point spread is a complicated and elaborate process that I hope to dissect before the next 231 days have expired. An understanding of probability will decrease your chances of leaving a casino with your wallet significantly lighter. Most importantly for me, probability also presents a number of quirky snig-bits of momentarily interesting trivia. My specialty. 

Imagine you walk into a room with 22 other people. You sit down and begin chatting, and for whatever reason the topic of birthdays pops up. What do you think the odds are that two people in that room will share the same birthday? 23/365 (or about 6.3%)? Less than that? Maybe more and less at the same time because math is a soulless bitch who wants nothing more than to mess with your head? 

Actually, the correct answer is about 50%. 

That’s right. In any gathering of 23 people, you’ve got a 50/50 shot of finding two people who would have to split the candle-blowing duties in the break-room when the staff gathers to sing off-key. If you’re wondering how that can be, how the odds can go from 1 in 365 that a single person shares your birthday to a virtual coin-toss with only 23 people you aren’t alone. This is known as the birthday paradox, not because it goes against any set laws of physics or reason, but because it’s completely counter-intuitive. To take it a step further, in a room of 70 people the probability rockets up to 99.9%. 

The reason for our confusion is purely narcissistic. We look at that room with 22 strangers and see 22 possibilities that someone else has our birthday – not much chance. But we need to add up all the possibilities between everyone: 22 for you, plus 21 for the next guy (not counting matching up with you), plus 20 for the next, and so on, giving us a total of 253 pair combinations. Of all those pairs, it’s likely (or at least 50.05% likely) that one of them will involve a match. There is actual math stuff behind these calculations, but I don’t want to dig much deeper because the math looks like this: 

It’s best just to take my word for it. Or try it out here – this page lets you run the a sample set of any size and keeps track of the matches for you. I tried it with 23 people and found a match 15 out of 25 times, so 60%. 

Let’s look at another dollop of probable goofiness. Consider these two scenarios: 

1. Mr. Brimley has two kids. The older child is a girl. What is the likelihood that both kids are girls? 

1. Mr. Ameche has two kids. At least one of them is a boy. What is the likelihood that both kids are boys? 

Common sense – or at least the suspicious breed of common sense that wears the logic home jerseys in my brain – tells me that the answer to both is 50%. Both problems assign a gender to one kid, so the other kid has a 50/50 chance of being a boy or girl. But if that were the case, there’d be nothing to write about, except maybe that it’s fun to substitute names from Cocoon movie actors into math problems. 

To break this down, we have to first understand that there are four permutations of the answer: BB, BG, GB, GG. (‘B’ means boy and ‘G’ means girl, and the one listed first is the older one.) In the case of Mr. Brimley, we know the older child is a girl, so that eliminates BB and BG. Thus, of the two remaining possibilities (GB and GG), there is indeed a 50% chance that both kids are girls. 

The second question is where the fun is hiding. At least one kid is a boy, which only eliminates GG. That leaves three other permutations (BB, BG and GB), and a 1 in 3 chance that both kids are boys. It’s all in the phrasing. 

At this point your math-inclined brain-matter is either shrugging its goopy grey shoulders because this is way beneath your level, or it’s exploding from these fantastic probability riddles. Or perhaps you still don’t understand – that’s okay, this is the best I can explain it. In high school I excelled at film studies and cafeteria three-card monte, so this is a smidgen over my head too. 

Except that with anything and everything in the real-life, reach-out-and-graspable, smash-it-with-a-hammer world, there are too many variables to kick dust in the face of mathematical probability, no matter how dumbed-down it is on paper. The math of the birthday paradox assumes 365 days (no leap years), and that there is an equal chance of people being born on any given day. The truth is that more people are born in the spring, more hospitals schedule C-sections or inducements on Mondays and Tuesdays, and identical twins can muck up the balance. Nothing is simple. 

The boy or girl paradox is no better. First off, there is a slightly greater likelihood of a male being born – that’s just biology. Genetics play into it also. Identical twins (which are always – well, almost always – the same gender) fiddle with the balance of this problem again. And how do you account for an intersex child? Like I said, real-life always soils the paper upon which a good math problem is sprawled. 

Maybe that’s my problem with math: I just don’t trust it. That’s okay, the probability that I take another math class in my lifetime is safely tucked against the zero mark. I’m good with that.