The Birthday Paradox

Hey, what’s up? It’s Matt and have you ever heard of the birthday paradox It’s the idea that if you have 23 people in a room together the odds that at least two of them share a birthday are Greater than 50% I remember when I first heard this. I was like no way that can’t be right There’s no way with 365 potential birthdays that only 23 people guarantees you a greater than 50% Chances there being a repeat no way even the name of the problem the birthday paradox Reflects the fact that it’s so counterintuitive for so many people so let’s look at this problem together and see how unbelievably, it’s actually true instead of starting with all 23 people and finding the odds that at least two of them share a birthday we’re actually
gonna Take this step by step, person by person. and see the odds that no one shares any birthday For example if person one walks into an empty room the odds that his birthday will be unique meaning it won’t match with anyone elses are 365 out of 365 because no one else is in the room But when the second person walks in the odds that his birthday will be unique are 364 out of 365 because person one’s birthday could match. When a third person walks in the room the odds that his birthday is unique are 363 out of 365. And this pattern continues as more people come into the room Assuming no birthdays have overlapped yet now these here are the odds per Individual that they don’t share a birthday with those already in the room, but we want to find the odds that ALL the individuals; Everyone in the room doesn’t share a birthday. To find that we multiply all the individual odds together This gives us the probability that no one shares a birthday So to find the probability that at least one birthday is shared we subtract this number from one. Now with five people in the room The odds of this happening are pretty low But this number increases really quickly as more people are added: 10 people bring the odds to 12 percent 15 people bring the odds to 25% and 20 people bring the odds of at least one shared birthday to 41% Finally 23 people bring the odds that there are no repeat birthdays to 49.27% Which means the odds that there is at least one repeated birthday are 50.73%. And as you keep adding people to this room the odds that at least one birthday is shared still keep increasing at a fast pace After 23 people gives us 50% it only takes about 20 more to get us into the 90s and by 70 People there is an incredibly small chance that a birthday isn’t shared by at least two people. Now you’ve seen the mathematical information (the proof) that shows you this is real, but it still feels weird right? Even though you’ve seen the math It’s doesn’t feel right that only 23 people gave a greater than 50% chance of their being at least one repeated birthday? well You’re not alone and this actually relates to a criticism of the birthday problem Levied by mathematician Paul hamos. “The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation What calculators do not yield is understanding or mathematical facility or a solid base for more advanced generalized theories” or; Said another way “people are focusing so much on the math of the problem, they don’t understand why it makes sense.” And he’s right. I explained a very mathematical mechanical way of viewing this problem and while it’s accurate that explanation doesn’t help people understand conceptually what’s going on It’s easy to say “let’s multiply some numbers together”, but what does that actually mean? The key to understanding, actually understanding, this paradox Is that there are many more birthday comparisons and therefore potential repeats than we realize. See, when we hear this problem, we typically think of only one set of comparisons. “What’s the likelihood someone else was born on this date?” If there are only 23 people then there are only 22 chances for this to happen. But the key piece to understanding this entire problem is that while there are only 22 chances for 1 specific birthday to repeat We’re looking for the probability that ANYONE’S birthday repeats Not just any specific one and when we understand this it becomes obvious that there are way more than 22 comparisons. When we consider all possible combinations of 23 birthdays, we actually have 253 comparisons. 253 chances to find a repeat birthday. And do you remember what the odds were that the second person walking into our room would have a unique birthday? It was 364 out of 365 Those are the odds of any two people not sharing a birthday Well when we raise it to the power of the number of comparisons we ultimately make, we get the same answer as before. So, there you go. You now know the two reasons why this paradox is Actually Accurate. There’s the unintuitive mathematical way we talked about first, But you should also now understand the conceptual reasoning behind it. This occurs because we are able to make way more xomparisons than there are people in the room, and it doesn’t matter whose birthday is repeated as long as 1 person’s is. Aaaand that’s it! Thanks for watching If you enjoyed this I made a second video on my other channel Everything else. You can click up here-ish to check that out. It’s basically using some of the same methods we talked about in this video, to solve other related types of math problems, and I thought it was cool. So check it out if you want to. Otherwise, that’s it. I’ll see you later. Thanks. Bye! (Subs by NotaNerdfighter)