As the number of people brought into the group increases, the chance of there being no match decreases. By the time the crowd numbers 23 people, the probability of no matching birthdays is .4927. Thus, the chance of at least one match within a group of 23 people is .5073, or slightly better than 50 percent. The reason the number is as low as 23 is that you aren't looking for a specific match. It doesn't have to be two particular people or a given date.
Any match involving any date or any two people is enough to create a coincidence.
Indeed, there are 253 different pairings possible among 23 people, any of which could lead to a match. Variations of the birthday problem serve as useful models for analyzing coincidences, says statistician Persi Diaconis of Stanford University. It's possible to derive a simple formula that gives an approximate answer for the number required to get a 50-percent chance of a match. That number is 1.2 times the square root of the number of categories. To get a 95-percent chance of a match, multiply the square root of the number of categories by 2.5.