Stuff You Should Know - Short Stuff: Birthday Paradox

Episode Date: December 7, 2022

The Birthday Paradox involves math, so you know this one will go perfectly.See omnystudio.com/listener for privacy information.

I'm Munga Shatikler and it turns out astrology is way more widespread than any of us want to believe. You can find it in Major League Baseball, international banks, K-pop groups, even the White House. But just when I thought I had a handle on this subject, something completely unbelievable happened to me and my whole view on astrology changed. Whether you're a skeptic or a believer, give me a few minutes because I think your ideas are about to change too.

Welcome to Skyline Drive on the iHeart Radio app, Apple Podcast, or wherever you get your podcasts. 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. Yes. Why would you do this to us?

Why? Is this math? Yes. It's not just math, it's famously incomprehensible math and we're going to talk about it and explain it. So thank you for that. Yeah, I will say that Laurie Eldove from HouseToForks.com did a pretty good job of explaining it I think.

But I picked this because let me tell you a little story real quick, since it's short stuff. Let's eat up some time. Flashback to seven and a half years ago. Okay, allow me. When 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, geez, 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 of it. I knew anyone personally. And I went to celebritybirthdays.com or whatever the website is, and I saw your face and two things happen, three things happen. The first thing that happened was, you got to 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 celebritybirthdays.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 clout I might have at famousbirthdays.com and nominated 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. Oh, that could be even more embarrassing and more egg on my face if they go, no. No. You deserve it. I even said, I was like, he's at least as famous as I am if I'm on there, he should be on there too. So it just seems right, you know? So you guys share a birthday, which is very cool and awesome and fun.

And I just think it's lovely now, even though I was 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. You're 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 365? Yeah. And actually, I think if you 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 364 chance, I think, if you put two people in a room together. The thing is, 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 a room, it's not like there's a three in 364 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 normies are not very good at calculating things exponentially 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 a birthday. Right.

And they're, 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 in and explain what the heck is going on here. How about that? Let's do it. I'm Mangesh Atikular and to be honest, I don't believe in astrology, but from the moment I was born, it's been a part of my life in India.

It's like smoking. You might not smoke, but you're going to get secondhand astrology. And lately I've been wondering if the universe has been trying to tell me to stop running and pay attention because maybe there is magic in the stars if you're willing to look for it. So I rounded up some friends and we dove in and let me tell you, it got weird fast. Tantric curses, Major League Baseball teams, canceled marriages, K-pop. But just when I thought I had to handle on this sweet and curious show about astrology,

my whole world came crashing down. It doesn't look good, there is risk to father. And my whole view on astrology, it changed. Whether you're a skeptic or a believer, I think your ideas are going to change too. Listen to Skyline Drive and the I Heart Radio App, Apple Podcast, or wherever you get your podcasts. 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 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? 23. Yeah, that's right. Wow, did you do that off of your dome? No, that's the answer. The larger the group you have, the greater the odds are, obviously.

So 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've got your 23 people. And if you're comparing just yourself to the other 22 people in the room, then you're just going to end up with those 22 comparisons. So 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 22 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 misguides us in this case because we fail to think about all the other people who connect with other people, right? That's right.

So I've seen a couple of ways to do this. One way is to say that if you have 23 people in a room, you have 23 people times 22 possible pairings, divide that number by two and what you end up with is 253. 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 20 possible pairings, right?

Because if you connect each person one time, you're going to come up with 20 possible pairings, but half of those are redundant, right? So connecting A to B in person B to A is the same thing. That's why you divide that number by two, right? So you got five times four equals 20 divided by two, which means you have 10 genuinely possible pairings in total. Another way to do it to get to the number is you take the first comparison, 364 to 365 divided by 365 and then for the next person, 363 divided by 365 and the next person, 362

and so on and so forth. If you do that for 23 different people and you take each of the products of those equations, all those little tiny percentages and multiply them, what you come up with is 49.83%, which means that what you've just done is show that there is a 49.83% that they're not going to have a birthday and then you just figure out the inverse of that. You come up with a 50.17% chance with 23 people that two of them are going to share the same birthday. Again, it's because you're not coming up with 23 comparisons, there's 253 comparisons and

of just 365 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 make those 22 comparisons that first person does and then the next person makes 21 comparisons, the next person makes 19, again because they've already made those other comparisons. All you do is add those numbers all up, 1918, so on and so on. Adding those together will eventually lead to those 253 comparisons or combinations of

comparisons rather. There's something that escapes me. We just generally explained it well, though I'm sure there are some people out there cringing, laughing, crying, who know about this kind of stuff. This is just the saddest thing I've ever heard. We generally explained it. I still don't understand how 253 comparisons for a possible pool of 365 dates leads to a 50% chance or 23 people.

Were you looking for an answer for me? It doesn't make sense. Okay. I'm just airing a grievance. Yeah. Really more than anything. I don't understand it at all. The upshot of it though is that when you get to 70 people, the pairings have grown so

exponentially that there's a 99, greater than a 99.9% chance that there will be a pair of people that share a birthday. Again though, we're talking about more than 2,000 comparisons for 365 days. Why is that not like 500% chance that there's going to be two people that have the same birthday? Yeah. I don't know. Another kind of cool thing that Laurie from the How Stuff Works article included, which

is just another kind of fun example of how exponential growth works is, and this is I think, 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. If you think of it in terms of money, the example that he used is 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 and so on for 30 days, it might not seem like much money, but at the end of the 30th day, that is $10.7 million. Right. Right. Millionaires who are good at math love to do that to people because they turn down this good deal and then they explain to them how it was a great deal and they're so dumb. Yeah. That's how the robber barons hoodwinked a generation of people.

That's right. Can we please end this torment? Yeah, I'm done. Okay. Short Stuff is out. Stuff You Should Know is a production of iHeartRadio. For more podcasts on iHeartRadio, visit the iHeartRadio app. Podcasts are wherever you listen to your favorite shows.

I'm Munga Shatikular and it turns out astrology is way more widespread than any of us want to believe. You can find it in Major League Baseball, international banks, K-pop groups, even the White House. But just when I thought I had a handle on this subject, something completely unbelievable happened to me and my whole view on astrology changed. Whether you're a skeptic or a believer, give me a few minutes because I think your ideas are about to change too.

Listen to Skyline Drive on the iHeartRadio app, Apple Podcasts, or wherever you get your podcasts.