The Birthday Paradox involves math, so you know this one will go perfectly.
Hey, and welcome to the Short Stuff. I'm Josh, and there's Chuck. Jerry's hanging out and Dave is here in spirit as always and this is short stuff. And Chuck, I have a question for you about this one. Why would you do this to us? Why it gets this math? Yes, it's not just math. It's famously incomprehensible math. I'm going to talk about it and explain it, so thank you for that. Yeah, I will say that uh Lori L. Dove from house stuff works dot com did a pretty good job of of explaining it, I think. Um, but I picked this because let me tell you a little story real quick, since it's short stuff. Flashback to um seven and a half years ago. Okay, allow me when uh Emily and I were waiting on our daughter to be born. We were adopting her and just she was late and late and late, and I was like, jeez, when is this kid going to come out? And finally when she was born, I was like, oh, I'm curious what celebrities she shares a birthday with. And you know, I had a lot in my mind at the time, so I wasn't thinking if it I knew anyone personally and I went to Celebrity Birthdays dot com or whatever the website is, and I saw your face, and two things happen. Three things happen. Uh. The first thing that happened was you gotta be kidding me seriously. The second thing that happened was, oh, that's actually really great because I'll never forget Josh's birthday like my whole life, and it's kind of cool that you guys share a birthday. And then the third thing was what is Josh doing on Celebrity Birthdays dot com? And why am I not on it? Well, my friend, I have an update for you because you told that story not too long ago and it got me into action. So I used whatever cloud I might have at Famous Birthdays dot com and um dominated you to be on the site as well. So hopefully I'm hoping that they will listen and then get you up there on before your birthday. So that could be even more embarrassing and more egg on my face if they go, no, you deserve it. I even said, I was like, he's at least as famous as I am. If I'm from there, he should be on there too, So it just seems right, you know. So you guys share birthday, which is very cool and awesome and fun, and I just think it's lovely. Now. Even though it's initially like what because you don't want to, like I don't know something about sharing birthdays, some people can get a little selfish, be like I want my birthday to myself. But what we're talking about is sharing birthdays. And what are the odds of sharing birthdays with someone? You would think it would be one in three d and sixty Yeah, And actually I think if you, um put two people in a room together, that is the odd although I'm sure I'm wrong about it right out of the gate. No, I am wrong. I was. There's a one in three hundred and sixty four chance if you put two people in a room together. The thing is, um, if you start putting more people in the room together, the chances don't increase linearly. It's not if you put three people in the room. It's not like there's a three in three hundred and sixty four chance. Man, math, It's not like it just increases linearly, like one after the other after the other. It starts to increase exponentially, and what you end up with is what's called the birthday paradox, which if you say that to anyone who knows anything about math, they will laugh at you and say, it's not actually a paradox. It's just that most people don't understand it. We really call it the birthday problem. Yeah, because here's the thing. And the more you read about this, and the more mathematicians you talk to, they all kind of very like they kind of pat you on the head and laugh a little bit and say, oh, you norms are not very good at calculating things exponentially correct like we are. We are very good at it because we have studied it and trained to do so. But you people, just your little p brains just don't think that way, and so you do very rudimentary math that is completely wrong when it comes to figuring out like the odds of sharing or odds of a lot of things, but the odds of sharing your birthday, right and there. It's true. They don't have to say it, but it is true. It is true. I say, we take a break and then we come back and explain what the heck is going on here? How about that? Let's do it, okay, Chuck, So we should set up the birthday problem or birthday paradox to you and me, The question is this, how large is a group of people, random people where every day of the year, excluding leap years, has an equal chance of, um being somebody's birthday, and there are no twins, it's all individual people. How many people do you have to get in the group before two of them will share a birthday? That's right? Wow, did you do that off of off of your dome? Know that that's the answer. The larger the group you have, the greater the odds are obviously, um. So it Yeah, it's an exponential math problem, and our brains don't generally think that way. So what you have to do is you have to look at the number of people in a room. Let's say you got your twenty three people, and if you're comparing just yourself to the under other twenty two people in the room, then you're just gonna end up with those twenty two comparisons. But when you're talking about exponential math, you can't just look at the one person in that room. You have to compare that probability for all the people in the room. So the first person would say, all right, I have those twenty two comparisons. Then the next person would step up and do the comparison, but there would be one less because they've already been compared to the one first person. And so on and so on until you get to the last person. Yeah. Our syllopsism uh misguides us in this case because we fail to think about all the other people who connect with other people. Right. So I've seen a couple of ways to do this. One way is to say, um that if you have twenty three people in a in a room, Um, you have twenty three people times twenty two um possible pairings. Divide that number by two, and what you end up with is two and fifty three. Okay, that's a really simple easy way that Ted Ed taught me to do it. But you have to get to the number. Let me put it in a different way, Chuck. For that formula, Let's say you have five people. Five people have UM twenty possible pairings, right, Okay, because if you if you connect each person one time, you're gonna come up with twenty possible pairings. But half of those are redundant. Right, So connecting A to B in person B two A is the same thing. That's why you divide that number by two, right, so you got five times four equals twenty divided by two, which means you have ten genuinely possible pairings in total. Another way to do it, to get to the number is you take that one the first comparison, three hundred and sixty four to three hundred and sixty five divided by three sixty five, and then for the next person, three hundred sixty three divided by three sixty five, and the next person three sixty two, and so on and so forth. And if you do that for twenty three different people and you take each to the products of those equations, all those little little tiny percentages, and multiply them, what you come up with is forty nine point eight three percent, which means that what you've just done is show that there is a forty nine point three percent that they're not going to have a birthday. And then you just figure out the inverse of that, and you come up with a fifty point seventeen percent chance with twenty three people that um two of them are going to share the same birthday. And again it's because you're not coming up with twenty three comparisons, there's two hundred and fifty three comparisons, and of just three hundred and sixty five days in a year. That's right, I guess. The last part of because there's sort of the third way to do it, which I kind of started but didn't even really finish, is you know, you make those twenty two comparisons that first person does, and then the next person makes twenty one comparisons, next person makes nineteen again because they've already made those other comparisons, and all you do is add those numbers all up, you know, so on on so on, and adding those together will eventually lead to those two hundred and fifty three comparisons or combinations of comparisons. Rather, so there's something that escapes me. We just generally explained it well. Although I'm sure there's some people out there cringing, laughing, crying, who who know about this kind of stuff. They're like, this is just the saddest thing I've ever heard. We generally explained it. I still don't understand how two hundred and fifty three comparisons for a possible pool of three hundred and sixty five dates leads to a fifty percent chance for three people. It doesn't make sense. I'm just airing a grievance really more than anything, I don't understand it at all. Um. The upshot of it, though, is that, um, when you get to seventy people, the pairings have grown so exponentially that, um, there's a ninet greater than a ninety nine percent chance that there will be a pair of people that share a birthday. Again, though we're talking about more than two thousand, um comparisons for three hundred and sixty five days. Why is that not like five percent chance that there's going to be two people that have the same birthday? Yeah, I don't know. Uh. With another kind of cool thing that was um that Laurie from the House Stuff Works article included, which is just another kind of fun example of how exponential growth works is and this is I think, um, I think she might have interviewed a mathematician. Yeah, his name is Frost. Oh yeah, he was laughing at you and me the whole time and he doesn't even know us. Yeah, he's the one that was like, yeah, you guys just aren't very good at this. Uh. Is if you're like, if you think of it in terms of money, and the example that he used is, um, if you're going to be offered a one penny on the first day, then two pennies on the second, three pennies on the third, and then so on so on for thirty days. It might not seem like much money, but at the end of the thirtieth day, that is ten point seven million dollars. Right. Millionaires who are good at math love to do that to people because they turned down this good deal, and then they explained to them how it was a great deal and they're so dumb. That's how the robber barons hoodwinked the generation of people. That's right. Uh, can we please end this torment? Yeah, I'm done, Okay, Choice stuff is out. Stuff you Should Know is a production of I Heart Radio. For more podcasts my heart Radio, visit the i heart Radio app, Apple Podcasts, or wherever you listen to your favorite shows.