Doesn't the "January 1" effect (more births in some months) change the answer?

Real birthdays aren't uniformly distributed, but this only increases the chance of collisions. The 23-person threshold is a worst-case estimate.

What about people born on February 29?

Including leap days drops the probability slightly (366 days instead of 365), but the change is invisible to two decimal places.

Is there a clean way to remember the formula?

In a domain of $D$ items, the 50% collision point is at about $\sqrt{2 D \ln 2} \approx 1.18 \sqrt{D}$.

Why do crypto people care about this?

Because finding a hash collision is often equivalent to forging a digital signature. If your hash output is 128 bits, an attacker needs only about $2^{64}$ work to find a collision.

How many people for someone to share my birthday?

About 253. The chance no one in $n$ people shares yours is $\left(\tfrac{364}{365}\right)^n$. Set that equal to 0.5 and solve.

Probability · Interactive

The Birthday Paradox

In a room of 23 people, the probability that two of them share a birthday is more than 50%. With 70 people it's 99.9%. Here is why, and a live slider you can drag from 1 to 100 to watch the probability curve climb.

Quick answer

In a room of 23 people, the chance two share a birthday is 50.7%. In a room of 50 it's 97.0%. In a room of 70 it's 99.9%. The trick is that the question isn't "does anyone share my birthday" (which would need ~253 people) but "do any two people share a birthday with each other." In a room of 23, there are $\binom{23}{2} = 253$ different pairs of people who could match.

Open this in math.im for the full interactive version →

Canonical URL: https://math.im/blog/birthday-paradox.html

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