What the numbers mean
A small room can contain a lot of possible pairs.
In a group of 23, there are 253 different pairs of people. Under this model, the chance that at least one pair shares a birthday is about 50.7%. With 22 people, it is still below half, at about 47.6%.
Read the sourcesCount pairs, not just people
If you look only for someone matching your birthday, there are 22 possible matches in a group of 23. But this question includes everyone matching everyone else. Count each pair once: 23 × 22 ÷ 2 = 253. The number of possible connections grows much faster than the size of the group. [1]
It is easier to count no matches
The first person can have any birthday. The next must avoid that day, leaving 364 out of 365 possibilities. The third must avoid two days, and so on. Multiplying those chances gives the probability of no shared birthdays. Subtracting it from one gives the chance of at least one match. [2]
More likely still leaves room for surprise
A 50.7% chance is far from a guarantee. Almost half of groups of 23 would have no match under this model. Real birthdays also vary by season, and leap day adds a complication. The model explains the pattern; it does not predict the birthdays in a particular room. [2]