Imagine you’re at a party. There are 23 people in the room, including you. Someone proposes a bet: “I bet at least two people here share the same birthday.”
You do the mental math. With 365 days in a year, that seems almost impossible. But when the odds are calculated, the result is astonishing: there’s a 50% chance two people in the room share a birthday. And if the room swells to 75 people? The odds rise to 99.9%.
This counterintuitive result, known as the Birthday Paradox, has delighted curious minds for years. How can such a small group produce such surprising odds?
A Party Trick of Probability
At first glance, the birthday paradox doesn’t feel plausible. With 365 days in a year, many people instinctively assume you’d need around 183 people—half the total days—for there to be a good chance of overlap. But in reality, we need to look at the combinations of any birthday possibilities, which drastically shifts the odds.
When asked about shared birthdays, most people unconsciously focus on whether someone else shares their birthday. But that’s not the question. The problem considers all possible pairings in the group—hundreds of comparisons even in small gatherings.
Think of it this way: you’re not just comparing one person’s birthday with everyone else’s, you’re comparing everyone’s birthday with everyone else’s. So if you have 23 people in the room, you’re not just comparing 1 person to the other 22; that’s just the first step. After this first step, you need to take the second person and compare them with the 21 people you haven’t compared them with. Then, take the 3rd person and compare them with the 20 others, and so on.
Let’s break it down.
- Imagine a room with just two people, A and B. The likelihood that they share a birthday is a mere 0.27%.
- Let’s add a third person (C), and suddenly there are three potential pairings, not two (A and B; A and C; B and C). The chance of a match now rises to 0.82%.
- If you add a fourth person (D), you have 6 winning pairings.
By the time you reach 23 people, the number of pairings jumps to 253, creating a 50.73% probability of at least one match. Why 50.73%?
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For that, we have to dive into a bit more math
How the Math Works
In case you’re still confused, don’t worry.
To decipher the math behind the paradox, we need to flip the question on its head. Instead of calculating all the ways people can share a birthday, it starts by asking how they might avoid sharing one.
If you have 23 people in the room, you have 253 birthday pairings. For the first pairing, there’s a 1 in 365 odd (0.27%) that they have the same birthday, and a 364 in 365 odd (99.73%) that they don’t. Do this on and on for all the combinations, 253 times.
This means you multiply 99.75% with itself 253 times. This number, mathematically written as (0.99726027)253, leads you to 49.95%. It sounds strange, but try it out yourself in a spreadsheet — even a very high probability, multiplied with itself numerous times, can bring the number down. That’s 49.95% that two people don’t share the same birthday, so 50.05% that any two people share the same birthday.
Now, this figure still isn’t exactly correct for our case because the simplified method we used above is more of a numerical approximation. The reason is that after we’ve done all possible combinations for the first person, the odds are shifted slightly.
- For the first person, the odds of a similar birthday are 1 in 365.
- For the second person, we’ve already taken the first person’s birthday out of the pool, so the odds of a similar birthday are 1 in 364.
- For the third person, the odds of a similar birthday are 1 in 363.
When you apply this correction, the odds of two people in a group of 23 having the same birthday is 50.7297%.
We won’t dive into the formal mathematics behind this, but if you really want a proper summary, here it is:
The Probabilities Change Surprisingly
As more people are added to the room, the percentages ramp up sharply, as follows:
- At 30 people, it’s 70% likely you share a birthday with someone
- At 35 people, it’s 81% likely
- At 41 people, it’s 90% likely
- At 50 people, it’s 97% likely
- At 60 people, it’s 99.4% likely
- At 70 people, there’s less than a 1/1000 chance of not finding someone who shares your birthday
- At 100 people, it’s less than 1 in 3 million
- At 117 people, it is less than 1 in 1 billion
- At 133 people, it’s around 1 in 1 trillion
- At 148 people, it is 1 in 1.2 quadrillion.
- At 200 people, it’s 1 in half a nonillion (half a billion trillion trillion)
This is what the formula looks like for an equation that computes the probability that two people will share the same birthday out of n:
A Trick that Deceives our Brains
This principle applies to other scenarios as well, from cryptography to games of chance. In cryptography, a related concept, the “birthday attack,” exploits the same math to find hash function collisions.
The reason why most of us fail to notice these odds is because our brains naturally focus on linear relationships. People underestimate how quickly probabilities compound as group size grows. The number of possible pairings increases exponentially.
The birthday paradox is more than a party trick—it’s a lesson in the power of mathematics to shatter our intuitions. Next time you’re in a room with 23 strangers, try making a bet. Just don’t be surprised when the math is on your side.
