Birthday Paradox for the Facebook Era
I love probability. Simple, empowering, and pervasive in daily life, it is one of the truest exceptions to the age-old refrain “when are we ever going to use this”. However, one of the more fascinating aspects of probability to me is how vexingly counter-intuitive it can be at times. My favorite example of this is the birthday paradox (although the Monty Hall problem is a pretty close second).
If you’re not familiar with the birthday paradox, it’s actually quite straightforward. How many people do we need to gather in a room in order to guarantee that there is a 100% chance that two of them will have the same birthday? If we ignore leap years and the fact that birthdays are not uniformly distributed throughout the year, it’s actually pretty easy to figure this out using the pigeonhole principle. To wit, if we had 365 people in a room, there is a chance, infinitesimal as it may be, that each person has a unique birthday. Add one more person and suddenly there is no way to avoid having a duplicate birthday. The answer is 366.
Here is where things get interesting. How many people would we need in a room in order to guarantee at least a 50% chance that two of them would have the same birthday? Most people end up guessing exactly half of the previous answer, 183. The answer is, surprisingly, drastically lower — just 23.
Much of the reason why this answer seems so counter intuitive has to do with how poorly we cognitively grasp exponentials, specifically the tremendous compounding effect they can have.
By thinking of this problem not in terms of people, but instead in terms of pairs of people, we can start to rationalize the answer. Notice that the first person in the room can potentially share a birthday with 22 other people. And the second person in the room can share a birthday with 21 remaining others. And so on and so forth. Turns out, all those pairs start to add up.
Given 253 possible pairs of people, suddenly the fact that at least one of those pairs produces matching birthdays half the time doesn’t feel too crazy.
I woke up a few weeks ago groggy eyed, and not unlike most days, began doing the rounds on social media. Through the morning haze, I noticed that something seemed awry. Facebook normally makes it blindingly obvious which of my friends are celebrating their birthday that day, likely in some analytics-confirmed attempt to foster additional social interactions. But this morning, I couldn’t for the life of me find any sign of it.
After eschewing any thoughts of a yet another overnight Facebook redesign, I finally concluded that despite my legion of close friends, acquaintances long since forgotten, and my fair share of who-the-hell-is-thats, it was actually nobody’s birthday! What were the odds of that? At the time, I had just crossed the 1000 friend plateau. I started guessing: 1%? 0.1%? Well it turns out it’s pretty easy to calculate.
The result was actually strikingly higher than my intuition. This got me thinking. How often does this happen? How many times a year would there be no birthdays? Well that’s also pretty straightforward to figure out.
Strangely enough, that comes out to — yep, you guessed it, an expectation of 23 such days in the year.
For the rest of the day, something about the coincidence — it stuck with me. Buried within all of these equations lay a social truth about how drastically things have shifted in our world. This math problem which previously highlighted the perceived exceptionality of overlap, could now be flipped on its head to showcase the rarity of absence and isolation.
Facebook, Twitter, and all other incarnations of social media have entrenched themselves into our daily lives — and of course with good reason. But the flip side is that sometimes it can feel like we’re living our lives in a room filled with 1007 people, with it becoming increasingly difficult to filter out the signal from the noise.
So consider this a reminder, that while you’re inundated with the barrage of engagement announcements, ice bucket challenges, and other happenings of so many of those in the world, don’t forget to focus your energies on those who mean the most to you. Those dearest to you. You know, the ones you’d invite to your birthday.
