Most people answer 6 or 12, and they are answering a different question. They are quietly asking: does someone here share
my month? For one person that is 1 chance in 12, so it feels expensive. The question we actually asked is: does
any two people match? That changes everything, because pairs multiply much faster than people. Two people make 1 pair. Four people make 6. Six people make 15. With four of us there are six pairs, each with a 1-in-12 chance of sharing a month. You cannot simply add six 1-in-12 chances, because the pairs tangle together, but you are in the neighbourhood: 3 people give 24%, 4 people give 43%, 5 people give 62%, 6 people give 78%. By six people a shared month is the normal outcome, not the surprise.
The real-birthday version runs on exactly the same arithmetic, just with 365 days instead of 12 months. 23 people is enough for the odds to tip past half. That is one class, or a small office. 23 people, 253 pairs. And here is the part adults find hardest: in that same room of 23, the chance that somebody shares
your birthday is only about 6%. Two numbers, one room, different questions. We answer the question we feel, not the one that was asked.
That gap is the actual skill. Intuition counts people and remembers hits. Maths counts pairs and counts everything. A class of 30 children holds 435 pairs, so of course two of them share something. A street of 20 families has 190 pairs, so of course two have the same car. The coincidence is not the universe winking. It is arithmetic, and there are far more pairs around you than your gut bothers to count.
One honest caveat, because the tidy answer is slightly too tidy: real birthdays are not spread evenly across the year. Some months are busier, which nudges the true odds a little higher, not lower. So treat the numbers above as a floor, not a ceiling. The full maths, if an adult wants it:
Wikipedia — birthday problem.