Twenty-three: Birthday Probability

This is a school project to summerize Numberphiles videos.

Why is 23 important in soccer? It is the number of people on the pitch during a game. Two teams of eleven, and the referee.

Here is the question:

What is the probability that two of those people share a birthday?

We are not talking about the year, we are just talking about the date itself. Maybe it's the fourth of January, or the fifth of July. Now we are going to make the question a little bit easier, what is the probability that no one on that pitch shares a birthday. For your first player, it doesn't matter what birthday he has, but for your second player it does matter. What is the probability that your second player doesn't share a birthday? He will have out of the 365 days to choose from, a birthday on 364 out of 365 days (364/365) we are not including February 29th, and we are assuming that all days are equally likely. Your third player will have a choice of 363 out of 365. So far our equation has 364/365 x 363/365. The fourth player will have 362 out of 365 days. You can keep going... until your 23rd player, how many choices does he have? Well, he has 343 out of 365, so this is the number of choices he has for a birthday because we are looking at everybody not sharing a birthday. If you want to find the probability that no one shares a birthday, you will have to multiply 364/365 x 363/365 x 362/365 x 361/365 x 360/365 x 359/365 x 358/365 x 357/365 x 356/365 x 355/365 x 354/365 x 353/365 x 352/365 x 351/365 x 350/365 x 349/365 x 348/365 x 347/365 x 347/365 x 346/365 x 345/365 x 344/365 x 343/365!(not factorial just exclamation mark). So if you  multiply all those numbers out you will get a number and that number is right about 0.493 and if you aren't happy with probability like that, that is 49.3%, just slightly under half. But we are interested in the opposite question, what is the probability  that two people share a birthday. That is the opposite thing that we worked out, so the probability that someone does share a birthday is 0.507 or 50.7%, it is slightly over half! It is more likely for two players to share a birthday than for two players to not share a birthday. That is actually quite surprising. But if you think of it this way, think of all the pairs of people that you can make with 23 people, there are 253 pairs of people that you can make. Think of it that way and you start to see why it is quite likely that two of those people will share a birthday.