Odds of 2 people having same birthday

If just 23 people are in a room, there's a better-than-even chance at least two of them have the same birthday.

If two people are in a room together, then there's a 364/365 chance they do not have the same birthday (if we ignore leap years and assume that all birthdays are equally likely), since there are 364 days that are different from the first person's birthday that can then be the second person's birthday.

If there are three people in the room, then the probability that they all have different birthdays is 364/365 x 363/365: As above, once we know the first person's birthday, there are 364 choices of a different birthday for the second person, and this leaves 363 choices for the third person's birthday that are different from those two.

Continuing in this fashion, once you hit 23 people, the probability that all 23 have different birthdays drops below 50%, and so the probability that at least two have the same birthday is better than even.