If you take a group of people born in a non leap year you would need 366 people for a 100% chance that someone shares a birthday but only 23 people for a 50% chance that somebody shares a birthday?
In: Mathematics
Okay, so let’s take 23 people in a room and line them up, giving each one of them a number.
Person 1 is then going to compare their birthday to person 2, then person 3, and so on, all the way to person 23. That’s 22 comparisons.
Person 2 is then going to compare their birthday to everyone else in the line except for person 1 (because they already compared, they don’t need to again). That’s 21 more comparisons.
Person 3 will compare to everyone except 1 & 2, for 20 more comparisons. And you keep on going down the line until 22 and 23 compare birthdays.
All in all, you’re going to have 22 + 21 + 20 + 19…..+ 1 comparisons, a total of 253 comparisons.
Each one of those comparisons is going to have a 1/365 chance of having the same birthday. Logically, that also means that each one of those comparisons will have a 364/365 (or about 99.7%) chance of NOT having the same birthday. If you do something with a 99.7% chance of failing enough times in a row, eventually it’s going to succeed.
In this case, we can compute the odds by taking 364/365 and raising it to the power of 253. That comes out to approximately 0.4995, which means that there is about a 50% chance that out of all of those comparisons, none of them will have a matching birthday. And as you add more and more people, that 50% will keep dropping to smaller and smaller chances. But it’s only a 0% chance once you have 366 people, because that would account for every single day of the year, plus one, so there is no possible way for there not to be a match.
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