2 min
How many people do you need in a room to share a birthday?
The answer is 23, not 183. Because the question is not "does anyone share *my* birthday" but "does any pair match" — and 23 people make 253 possible pairs.
Answer
Before you say 183 — the answer is 23 people.
Here is the key. The question is not "does someone share my birthday". It is "do any two people in the room share a birthday". With 23 people you can form 253 pairs. Each individual pair has only a 1-in-365 chance of matching, which sounds like nothing. But 253 of those tiny chances stacked up is like holding 253 lottery tickets.
It is clearer if you work backwards. The second person has a 364/365 chance of missing the first person's birthday. The third has a 363/365 chance of missing both. Multiply all of those together and the chance that all 23 people have different birthdays is only about 49.3%. So the chance that at least one pair matches is 50.7% — just past half.
To see how big the gap is: to make "someone shares my birthday" more likely than not, you need 253 people. In the same room, one question needs 23 and the other needs 253 — eleven times as many. That gap is where the feeling of paradox comes from.
Mathematically, the chance that n people all have different birthdays is roughly e^(−n²/730). The chance of a match grows with the square of the number of people. At 70 people, it is already 99.9%.
In one line: the birthday problem is not really counter-intuitive. It just reminds us that "some pair matches" and "someone matches me" are two completely different questions.
Michael's Thought
23 people and it is more likely than not? My first reaction was that it cannot be true. My intuition was quietly answering a different question.
Sources
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