The Facts on The Birthday Paradox
Sometimes math can be fun, and sometimes it can explain things that seemed weird before. Based on probability, if there are at least 60 readers here, then there’s a 99% chance that two of you are having a birthday today. But there are 365 days in the year so how could 60 people virtually guarantee a shared birthday?
Today we’re talking about the birthday problem! Although it’s not really a problem unless you hate birthdays.
“It’s delicious cake but I do hate this day so much”
When A Paradox is Not a Paradox
The birthday problem is sometimes called the “Birthday Paradox” but it’s technically not a paradox. A paradox is a self-contradicting statement or thought, probably the most famous one is “what happens when an unstoppable force meets an immovable object.”
Although nowadays because of WandaVision, “The Ship of Theseus” is probably the most famous one at the moment, we even have an article on it that questions whether your body is like the eponymous ship, “Are You the Ship of Theseus”.
Back to the birthday problem, it’s actually what's called a veridical paradox, something that seems impossible but is actually possible like doing the dishes.
“I wash and wash and wash but everyday there are more dishes!”
In this case, the birthday problem states that in a room of 23 individuals there is a little over a 50% chance that two people have the same birthday. This is not including leap days and assumes an even distribution of birthdays.
The more people in the room the higher the probability that someone will share a birthday. Once you get to 60 people that’s a little over 99% chance two people will have the same birthday.
When I first heard this years ago, I just remember thinking they were nuts. That’s not how math works. So I dropped that college statistics class. Eventually, I realized I was wrong but you really can’t blame me for that.
How It Works
Now I know the things I said before, “a 50% chance of 2 people in a group of 23” sharing a birthday sounds blatantly ridiculous but here’s how it works, or at least what helped me understand it.
If you meet one stranger the chances of you two sharing a birthday is 1/365 or a .3% chance. The chances you don’t share a birthday is just the opposite, 364/365 or a 99.7% chance you won’t share a birthday.
If you enter a room with 22 strangers that means you have 22 different opportunities to have a matching birthday. Now the next person in that group, let’s call him Moe, gets to find out if they share a birthday with anyone, they already know they don’t share a birthday with you so their opportunities are the number of strangers they haven’t met, so 21.
Once Moe is done, the next person, Larry does the same and since they already met you and Moe, Larry only has to meet another 20 people.
This keeps going until everyone has met everyone else and compared birthdays. Add up all the opportunities each person had, you had 22 opportunities, plus Moe’s 21, plus Larry’s 20, plus the next person’s 19, etcetera. In the end, that means 23 people had 253 different opportunities to match a birthday.
The best way to do the next step is to find out the probability that none of the 253 opportunities had a matching birthday. You do this by taking the percentage of two people not matching (99.7%) and raising it by the number of total opportunities, in this case, 253. Giving you 49.995% chance of no one’s birthday matching, meaning there's a 50.005% chance that someone’s birthday matches.
There you go. At 23 people you have a 50% chance there will be a matching birthday. And if you explain all of this to someone at a party there’s about a 100% chance they will say “Happy Birthday” to you just to get away from you.
“Holy shit, I get it, Happy Birthday. Now leave me alone.”
Of course, as I said before, this assumes equal distribution of birthdays throughout the year and we know that’s not the case.
Data journalist Matt Stiles compiled data from the years 1994-2014 and found September is the most popular month for birthdays with Christmas and New Year’s Day being the two least popular birthdays.
I also somehow ended up on an article, which I have not fact-checked yet, that states babies born in October are the most likely to live to 100 years old. So in about 70 years, I’ll be able to confirm that and I’ll write a new article when I do.
Enjoy your 99-year birthday limit losers!
