Originally Posted by BigGreen
Starting where math and common sense actually agree, if two people are in the same room, the chance of them sharing the same birth date and month (just referred to as 'birthday' from here on in) is 1/365. The inverse truth (or whatever the hell they call it) would then say that chances of them not sharing a birthday are 364/365. Now when we toss a third person (and presumably a third different birthday) into the mix, the probability of that third bday being different is 363/365. The probability of the three birthdays being different is: (364/365) times (363/365), which gives you roughly 0.9917...so there is a 99.2% chance that three people have DIFFERENT bdays. Swap that inverse truth around and you get a 0.8% chance that the three random people SHARE a birth date and month. If you plug that probability formula into excel, or take the time to do it by calculator and paper, you can fill out the chart, and it's at exactly 23 people that the odds finally slip to UNDER 50% that they all have different birthdays, and thus OVER 50% that they do.
With even a random sampling of 50 people, there is a 97% chance that two share a birthday.
When you think about the 23 people as pairs instead of people, it makes more sense. There's what, 250 plus possible pairs that are generated from 23 people...so when you look at it in terms not of 23 people sharing a birthday, but of the odds of 250 pairs of people sharing a birthday, it makes a little more intuitive sense.
This is actually used in math courses to explain compound probability and i think it's called the soccer game birthday dilemma or something like that (since there are 23 people on the field including the ref at the beginning of the game when they shake hands...something like that). So if you were to google "soccer game birthday dilemma" i'm sure you'd get the FULL explanation.