Welcome to the party! Today you will participate in an interactive experiment about the birthday paradox. We will use you, the reader, as part of our data, to help explain what it is, why it is cool, and how it works.

Hello! I’m Russell. On what day of the year were you born? Please tell me your real birthday, this is for science 🔬not evil 💀, I promise.

Cool, thanks! My birthday is November 15th. We don’t share the same birthday 😢We share the same birthday! Scorpio fam 🦂.But how many people do you think we need in a room to have a 50% chance that two people have the same birthday?But let’s pretend I was lying, and I’m really born on, say... August 15th. How many people do you think we need in a room to have a 50% chance that two people have the same birthday?

This seems like a good guess and sounds intuitive💡because there are 365.2422 days^{+We
stay in sync with the seasons by adding a leap year every four years since it is close enough to
¼.} in the year, which is a lot. But people would really give us a
chance of a shared birthday. I would take that bet! 💸

Hold your horses 🐴! You know there are 365.2422 days^{+We stay in sync with the
seasons by adding a leap year every four years since it is close enough to ¼.} in a year,
right? The chances of a shared birthday with just
people is actually about . I wouldn’t take that bet! 💸

Oh so close 🥈! But people would really give us a chance of a shared birthday. Tough to say if I would take that bet... 💸

Holy 💩you nailed it! Or maybe you’ve heard about this, huh? Or you are just a big Michael Jordan 🏀 fan like myself… Anyways, it can still be hard to believe it actually happens, right?

The chance that two people in the same room have the same birthday — that is the Birthday
Paradox 🎉. And according to fancy
math, there is a 50.7% chance when there are just 23 people^{+This is in a hypothetical world. In reality, people aren’t born evenly throughout the year,
and leap years are excluded. However, the numbers should still be pretty close. More on this in the
appendix.} in a room. It may seem surprising, but the logic is
much more simple than it appears. But first, let me convince you it happens.

Of course I can make you believe it. But we won’t look at formulas 🤓 or simulations 🤖, let’s invite the last 21 real people like you who just visited this site to our party. 🎈= shared birthday

💥 Boom! I told you it could happen.🥚Aww we laid a goose egg. But that is just one instance, and it would be wrong of me to stop there. Because each time we get 23 people together, it’s basically a coin flip that we get a shared birthday. The more we repeat this, the more balanced our results will be. That is the Law of Large Numbers in action.

Okay, let’s bring in 19 more groups of 23 people and see what happens. I have a feeling we will witness the magic of the paradox a few times 🤞. Hmm let’s speed this up. A little more... A lot more...

As you can see, we are converging towards the 50.7% success rate^{+Since
this is a live experiment, the actual success rate may vary from this hypothetical value. However, the
numbers should still be pretty close. More on this in the appendix.}. But wait, there’s
more! Let’s boost this sample size to include everyone who has ever visited this
site. We will put it on hyperspeed ⚡️so we aren’t here all day.

The odds can seem surprising because as self-involved humans 😍, we usually frame the situation by comparing just ourselves to 365 possible days. Our brains are more inclined to think about things linearly, but as we add more people to a room, the number of comparisons actually goes up quadratically 📈.

The odds can seem surprising because as self-involved humans 😍, we usually frame the situation by comparing just ourselves to 365 possible days. Our brains are more inclined to think about things linearly, but as we add more people to a room, the number of comparisons actually goes up quadratically 📈.

Aww you didn’t have to say that 😉. Hopefully this helped demystify The Birthday Paradox. Now get out there and shout random probabilities at a room full of strangers!

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### Average Reader Guess

Results of all reader guesses for how many people are required to have a 50/50 chance of a shared birthday.

### Reader Birthday Distribution

The distribution of the number of real birthdays on each day collected from readers.

### Probability of a Shared Birthday

The hypothetical probability (assuming evenly distributed birthdays) of a shared birthday based on number of people in a room.

January 1

December 31

**About:** The data for this story is collected from readers like
you, and is constantly updating. New results are compiled every few minutes. All math is based on the
hypothetical even distribution of birthdays. Design assist from Jan Diehm. Characters are adapted from this project. Original character art
credit: Aidan O’Donohue. Get in touch at
russell@pudding.cool.

The Pudding explains ideas debated in culture with visual essays.