Birthday Paradox

Smallest n such that a shared birthday is more than 50%

Birthdays are assumed to be uniformly distributed over 365 days. We want the smallest number of people n for which the probability that at least two people share a birthday is greater than 50%.

Final answer: n = 23

At n = 23, probability ≈ 50.73%

Interactive probability explorer

Move the slider to see how the probability changes as the group gets larger.

Number of people

P(at least one shared birthday)

50.73%

Choose n

1 23 60

No shared birthday 49.27%

At least one shared birthday 50.73%

How the probability grows

The chance rises quickly because each new person can potentially match many people already in the room.

Why we use the complement

It is easier to compute the probability that all birthdays are different, then subtract from 1.

P(no match) = 365/365 × 364/365 × 363/365 × ··· × (365 - n + 1)/365 P(at least one match) = 1 - P(no match)

n = 22

47.57%

Still below 50%

n = 23
50.73%
First value above 50%

Step 1. The first person can have any birthday, so the probability is 365/365.

Step 2. The second person must avoid the first person's birthday, so the probability is 364/365.

Step 3. The third person must avoid the first two birthdays, so the probability is 363/365.

Continue. For n people, multiply all these terms together.

Finally. Subtract from 1 to get the probability that there is at least one shared birthday.

Intuition: even though 23 is much smaller than 365, what matters is the number of pairs of people. With 23 people, there are 23 × 22 / 2 = 253 pairs, which creates many chances for a birthday collision.

Therefore, the smallest number of people for which the probability exceeds 50% is 23.