The Birthday Problem
Probability Theory’s Most Counterintuitive Lesson
By the time this article is released, it’ll be my 19th birthday.
So why don’t we speak about a famous paradox within probability theory: The Birthday Problem (or Birthday paradox)!
Let me ask you a question:
How many randomly selected people are needed for there to be a ≥50% chance that at least two share the same birthday?
Intuitive Answer
Well, a year has 365 days, so maybe we can divide that by two and get 182 or 183 people… WRONG.
What if I told you that we only need 23? Here’s the proof:
Proof
First, assume the year is not a leap year, birthdays are uniformly distributed (every birthday is equally likely) and we do not have twins or external dependencies.
What’s the probability that all birthdays are unique for a group of n people?:
Person 1: Can have any birthday (365/365 = 1)
Person 2 has to avoid Person 1’s birthday (364/365)
Person 3 has to avoid Person 1 and Person 2’s birthday (363/365)
…
Person n has to avoid all other birthdays ( [365 - (n - 1) ] / 365)
The probability of no shared birthdays is:
Which is equivalent to:
Where 365! is the factorial of 365: 1 × 2 × 3 × … × 364 × 365.
The probability of at least one match is the complement:
Since 365! is a giant number, waaay larger than the total amount of atoms in the observable universe, let’s approximate and visualize the probabilities:
All probabilites were approximated using Python.
The math shows that at least 23 randomly selected people are needed for there to be a probability greater than fifty per cent that at least two share the same birthday.
Obviously, the probability is 100% (meaning it happens almost surely) for any n > 365 (pigeonhole principle).
Why did we guess wrong?
The birthday paradox confounds us because of how human cognition works—specifically through the lens of Daniel Kahneman’s Dual System Theory1. Our minds operate using two systems:
System 1, which is fast, intuitive, and relies on mental shortcuts, and
System 2, which is slow, logical, and capable of deliberate calculation.
When faced with the birthday problem, System 1 dominates. It sees 23 people and 365 days and intuitively concludes that matches must be rare, because 23 seems insignificant compared to 365. This instinct stems from the representativeness heuristic, where we underestimate how quickly connections accumulate. We might picture a simple one-to-one comparison—like checking if any of 22 newcomers share Person A’s birthday—but this grossly undercounts the reality.
The mathematical truth reveals why System 1 fails here. With 23 people, there aren’t just 22 comparisons; there are 22 + 21 + 20 + … + 1 = 253 unique pairings. For no shared birthdays to occur, all 253 pairs must avoid matching—a scenario with a mere 49.95% chance, given each pair has a 364/365 (99.73%) probability of mismatching. Thus, the chance of at least one shared birthday tips to 50.05%. System 1 misses this combinatorial explosion because it doesn’t naturally account for exponential growth in connections. It’s like imagining a room where people only shake hands with one neighbor, when in reality, everyone is shaking hands with everyone else.
This gap between intuition and reality is why Kahneman’s theory matters: System 1’s heuristic thinking saves mental effort but often leads us astray in probability problems. Only by engaging System 2—the analytical mode that methodically calculates pairings and compounding probabilities—can we see why 23 people create a 50% match chance.
The Birthday paradox isn’t just a math quirk; it’s a vivid demonstration of how our brains prioritize speed over precision, and why trusting our gut can sometimes lead us wildly off course.
Kahneman, D. (2011). Thinking, fast and slow. macmillan.