Latest / Stuff You Should Know / Short Stuff: Birthday Paradox

Transcript

Short Stuff: Birthday Paradox · Stuff You Should Know

• 0:04
  Speaker 1: Hey, and welcome to the Short Stuff. I'm Josh, and
• 0:06
  Speaker 1: there's Chuck. Jerry's hanging out and Dave is here in
• 0:09
  Speaker 1: spirit as always and this is short stuff. And Chuck,
• 0:12
  Speaker 1: I have a question for you about this one. Why
• 0:15
  Speaker 1: would you do this to us? Why it gets this math? Yes,
• 0:20
  Speaker 1: it's not just math. It's famously incomprehensible math. I'm going
• 0:24
  Speaker 1: to talk about it and explain it, so thank you
• 0:26
  Speaker 1: for that. Yeah, I will say that uh Lori L.
• 0:30
  Speaker 1: Dove from house stuff works dot com did a pretty
• 0:32
  Speaker 1: good job of of explaining it, I think. Um, but
• 0:36
  Speaker 1: I picked this because let me tell you a little
• 0:38
  Speaker 1: story real quick, since it's short stuff. Flashback to um
• 0:45
  Speaker 1: seven and a half years ago. Okay, allow me when
• 0:51
  Speaker 1: uh Emily and I were waiting on our daughter to
• 0:53
  Speaker 1: be born. We were adopting her and just she was
• 0:57
  Speaker 1: late and late and late, and I was like, jeez,
• 1:00
  Speaker 1: when is this kid going to come out? And finally
• 1:03
  Speaker 1: when she was born, I was like, oh, I'm curious
• 1:06
  Speaker 1: what celebrities she shares a birthday with. And you know,
• 1:10
  Speaker 1: I had a lot in my mind at the time,
• 1:12
  Speaker 1: so I wasn't thinking if it I knew anyone personally
• 1:14
  Speaker 1: and I went to Celebrity Birthdays dot com or whatever
• 1:17
  Speaker 1: the website is, and I saw your face, and two
• 1:24
  Speaker 1: things happen. Three things happen. Uh. The first thing that
• 1:27
  Speaker 1: happened was you gotta be kidding me seriously. The second
• 1:31
  Speaker 1: thing that happened was, oh, that's actually really great because
• 1:35
  Speaker 1: I'll never forget Josh's birthday like my whole life, and
• 1:39
  Speaker 1: it's kind of cool that you guys share a birthday.
• 1:42
  Speaker 1: And then the third thing was what is Josh doing
• 1:44
  Speaker 1: on Celebrity Birthdays dot com? And why am I not
• 1:47
  Speaker 1: on it? Well, my friend, I have an update for
• 1:50
  Speaker 1: you because you told that story not too long ago
• 1:53
  Speaker 1: and it got me into action. So I used whatever
• 1:56
  Speaker 1: cloud I might have at Famous Birthdays dot com and
• 1:59
  Speaker 1: um dominated you to be on the site as well.
• 2:01
  Speaker 1: So hopefully I'm hoping that they will listen and then
• 2:05
  Speaker 1: get you up there on before your birthday. So that
• 2:08
  Speaker 1: could be even more embarrassing and more egg on my
• 2:10
  Speaker 1: face if they go, no, you deserve it. I even said,
• 2:14
  Speaker 1: I was like, he's at least as famous as I am.
• 2:16
  Speaker 1: If I'm from there, he should be on there too,
• 2:18
  Speaker 1: So it just seems right, you know. So you guys
• 2:20
  Speaker 1: share birthday, which is very cool and awesome and fun,
• 2:24
  Speaker 1: and I just think it's lovely. Now. Even though it's
• 2:26
  Speaker 1: initially like what because you don't want to, like I
• 2:30
  Speaker 1: don't know something about sharing birthdays, some people can get
• 2:33
  Speaker 1: a little selfish, be like I want my birthday to myself.
• 2:36
  Speaker 1: But what we're talking about is sharing birthdays. And what
• 2:39
  Speaker 1: are the odds of sharing birthdays with someone? You would
• 2:42
  Speaker 1: think it would be one in three d and sixty Yeah,
• 2:46
  Speaker 1: And actually I think if you, um put two people
• 2:49
  Speaker 1: in a room together, that is the odd although I'm
• 2:53
  Speaker 1: sure I'm wrong about it right out of the gate. No,
• 2:56
  Speaker 1: I am wrong. I was. There's a one in three
• 3:00
  Speaker 1: hundred and sixty four chance if you put two people
• 3:04
  Speaker 1: in a room together. The thing is, um, if you
• 3:07
  Speaker 1: start putting more people in the room together, the chances
• 3:11
  Speaker 1: don't increase linearly. It's not if you put three people
• 3:14
  Speaker 1: in the room. It's not like there's a three in
• 3:17
  Speaker 1: three hundred and sixty four chance. Man, math, It's not
• 3:23
  Speaker 1: like it just increases linearly, like one after the other
• 3:26
  Speaker 1: after the other. It starts to increase exponentially, and what
• 3:30
  Speaker 1: you end up with is what's called the birthday paradox,
• 3:33
  Speaker 1: which if you say that to anyone who knows anything
• 3:35
  Speaker 1: about math, they will laugh at you and say, it's
• 3:37
  Speaker 1: not actually a paradox. It's just that most people don't
• 3:39
  Speaker 1: understand it. We really call it the birthday problem. Yeah,
• 3:43
  Speaker 1: because here's the thing. And the more you read about this,
• 3:46
  Speaker 1: and the more mathematicians you talk to, they all kind
• 3:48
  Speaker 1: of very like they kind of pat you on the
• 3:50
  Speaker 1: head and laugh a little bit and say, oh, you
• 3:53
  Speaker 1: norms are not very good at calculating things exponentially correct
• 4:00
  Speaker 1: like we are. We are very good at it because
• 4:02
  Speaker 1: we have studied it and trained to do so. But
• 4:05
  Speaker 1: you people, just your little p brains just don't think
• 4:08
  Speaker 1: that way, and so you do very rudimentary math that
• 4:11
  Speaker 1: is completely wrong when it comes to figuring out like
• 4:14
  Speaker 1: the odds of sharing or odds of a lot of things,
• 4:16
  Speaker 1: but the odds of sharing your birthday, right and there.
• 4:19
  Speaker 1: It's true. They don't have to say it, but it
• 4:21
  Speaker 1: is true. It is true. I say, we take a
• 4:23
  Speaker 1: break and then we come back and explain what the
• 4:25
  Speaker 1: heck is going on here? How about that? Let's do it, okay, Chuck,
• 4:50
  Speaker 1: So we should set up the birthday problem or birthday
• 4:52
  Speaker 1: paradox to you and me, The question is this, how
• 4:56
  Speaker 1: large is a group of people, random people where every
• 5:00
  Speaker 1: day of the year, excluding leap years, has an equal
• 5:03
  Speaker 1: chance of, um being somebody's birthday, and there are no twins,
• 5:09
  Speaker 1: it's all individual people. How many people do you have
• 5:12
  Speaker 1: to get in the group before two of them will
• 5:16
  Speaker 1: share a birthday? That's right? Wow, did you do that
• 5:22
  Speaker 1: off of off of your dome? Know that that's the answer.
• 5:27
  Speaker 1: The larger the group you have, the greater the odds
• 5:29
  Speaker 1: are obviously, um. So it Yeah, it's an exponential math problem,
• 5:36
  Speaker 1: and our brains don't generally think that way. So what
• 5:39
  Speaker 1: you have to do is you have to look at
• 5:41
  Speaker 1: the number of people in a room. Let's say you
• 5:43
  Speaker 1: got your twenty three people, and if you're comparing just
• 5:47
  Speaker 1: yourself to the under other twenty two people in the room,
• 5:52
  Speaker 1: then you're just gonna end up with those twenty two comparisons.
• 5:56
  Speaker 1: But when you're talking about exponential math, you can't just
• 5:59
  Speaker 1: look at the one person in that room. You have
• 6:01
  Speaker 1: to compare that probability for all the people in the room.
• 6:05
  Speaker 1: So the first person would say, all right, I have
• 6:09
  Speaker 1: those twenty two comparisons. Then the next person would step
• 6:12
  Speaker 1: up and do the comparison, but there would be one
• 6:14
  Speaker 1: less because they've already been compared to the one first person.
• 6:18
  Speaker 1: And so on and so on until you get to
• 6:20
  Speaker 1: the last person. Yeah. Our syllopsism uh misguides us in
• 6:26
  Speaker 1: this case because we fail to think about all the
• 6:28
  Speaker 1: other people who connect with other people. Right. So I've
• 6:31
  Speaker 1: seen a couple of ways to do this. One way
• 6:35
  Speaker 1: is to say, um that if you have twenty three
• 6:38
  Speaker 1: people in a in a room, Um, you have twenty
• 6:43
  Speaker 1: three people times twenty two um possible pairings. Divide that
• 6:50
  Speaker 1: number by two, and what you end up with is
• 6:54
  Speaker 1: two and fifty three. Okay, that's a really simple easy
• 6:58
  Speaker 1: way that Ted Ed taught me to do it. But
• 7:01
  Speaker 1: you have to get to the number. Let me put
• 7:03
  Speaker 1: it in a different way, Chuck. For that formula, Let's
• 7:05
  Speaker 1: say you have five people. Five people have UM twenty
• 7:09
  Speaker 1: possible pairings, right, Okay, because if you if you connect
• 7:14
  Speaker 1: each person one time, you're gonna come up with twenty
• 7:18
  Speaker 1: possible pairings. But half of those are redundant. Right, So
• 7:22
  Speaker 1: connecting A to B in person B two A is
• 7:25
  Speaker 1: the same thing. That's why you divide that number by two, right,
• 7:30
  Speaker 1: so you got five times four equals twenty divided by two,
• 7:33
  Speaker 1: which means you have ten genuinely possible pairings in total.
• 7:39
  Speaker 1: Another way to do it, to get to the number
• 7:40
  Speaker 1: is you take that one the first comparison, three hundred
• 7:44
  Speaker 1: and sixty four to three hundred and sixty five divided
• 7:46
  Speaker 1: by three sixty five, and then for the next person,
• 7:49
  Speaker 1: three hundred sixty three divided by three sixty five, and
• 7:52
  Speaker 1: the next person three sixty two, and so on and
• 7:54
  Speaker 1: so forth. And if you do that for twenty three
• 7:57
  Speaker 1: different people and you take each to the products of
• 8:01
  Speaker 1: those equations, all those little little tiny percentages, and multiply them,
• 8:07
  Speaker 1: what you come up with is forty nine point eight
• 8:09
  Speaker 1: three percent, which means that what you've just done is
• 8:13
  Speaker 1: show that there is a forty nine point three percent
• 8:16
  Speaker 1: that they're not going to have a birthday. And then
• 8:18
  Speaker 1: you just figure out the inverse of that, and you
• 8:20
  Speaker 1: come up with a fifty point seventeen percent chance with
• 8:23
  Speaker 1: twenty three people that um two of them are going
• 8:26
  Speaker 1: to share the same birthday. And again it's because you're
• 8:28
  Speaker 1: not coming up with twenty three comparisons, there's two hundred
• 8:31
  Speaker 1: and fifty three comparisons, and of just three hundred and
• 8:35
  Speaker 1: sixty five days in a year. That's right, I guess.
• 8:40
  Speaker 1: The last part of because there's sort of the third
• 8:43
  Speaker 1: way to do it, which I kind of started but
• 8:44
  Speaker 1: didn't even really finish, is you know, you make those
• 8:47
  Speaker 1: twenty two comparisons that first person does, and then the
• 8:50
  Speaker 1: next person makes twenty one comparisons, next person makes nineteen
• 8:55
  Speaker 1: again because they've already made those other comparisons, and all
• 8:58
  Speaker 1: you do is add those numbers all up, you know,
• 9:01
  Speaker 1: so on on so on, and adding those together will
• 9:04
  Speaker 1: eventually lead to those two hundred and fifty three comparisons
• 9:08
  Speaker 1: or combinations of comparisons. Rather, so there's something that escapes me.
• 9:13
  Speaker 1: We just generally explained it well. Although I'm sure there's
• 9:16
  Speaker 1: some people out there cringing, laughing, crying, who who know
• 9:20
  Speaker 1: about this kind of stuff. They're like, this is just
• 9:22
  Speaker 1: the saddest thing I've ever heard. We generally explained it.
• 9:26
  Speaker 1: I still don't understand how two hundred and fifty three
• 9:31
  Speaker 1: comparisons for a possible pool of three hundred and sixty
• 9:36
  Speaker 1: five dates leads to a fifty percent chance for three people.
• 9:44
  Speaker 1: It doesn't make sense. I'm just airing a grievance really
• 9:47
  Speaker 1: more than anything, I don't understand it at all. Um.
• 9:50
  Speaker 1: The upshot of it, though, is that, um, when you
• 9:53
  Speaker 1: get to seventy people, the pairings have grown so exponentially that, um,
• 9:59
  Speaker 1: there's a ninet greater than a ninety nine percent chance
• 10:02
  Speaker 1: that there will be a pair of people that share
• 10:05
  Speaker 1: a birthday. Again, though we're talking about more than two thousand,
• 10:10
  Speaker 1: um comparisons for three hundred and sixty five days. Why
• 10:15
  Speaker 1: is that not like five percent chance that there's going
• 10:18
  Speaker 1: to be two people that have the same birthday? Yeah,
• 10:23
  Speaker 1: I don't know. Uh. With another kind of cool thing
• 10:28
  Speaker 1: that was um that Laurie from the House Stuff Works
• 10:30
  Speaker 1: article included, which is just another kind of fun example
• 10:33
  Speaker 1: of how exponential growth works is and this is I think, um,
• 10:38
  Speaker 1: I think she might have interviewed a mathematician. Yeah, his
• 10:41
  Speaker 1: name is Frost. Oh yeah, he was laughing at you
• 10:43
  Speaker 1: and me the whole time and he doesn't even know us. Yeah,
• 10:45
  Speaker 1: he's the one that was like, yeah, you guys just
• 10:47
  Speaker 1: aren't very good at this. Uh. Is if you're like,
• 10:50
  Speaker 1: if you think of it in terms of money, and
• 10:52
  Speaker 1: the example that he used is, um, if you're going
• 10:55
  Speaker 1: to be offered a one penny on the first day,
• 10:58
  Speaker 1: then two pennies on the second, three pennies on the third,
• 11:02
  Speaker 1: and then so on so on for thirty days. It
• 11:05
  Speaker 1: might not seem like much money, but at the end
• 11:07
  Speaker 1: of the thirtieth day, that is ten point seven million dollars. Right.
• 11:12
  Speaker 1: Millionaires who are good at math love to do that
• 11:14
  Speaker 1: to people because they turned down this good deal, and
• 11:16
  Speaker 1: then they explained to them how it was a great
• 11:18
  Speaker 1: deal and they're so dumb. That's how the robber barons
• 11:21
  Speaker 1: hoodwinked the generation of people. That's right. Uh, can we
• 11:25
  Speaker 1: please end this torment? Yeah, I'm done, Okay, Choice stuff
• 11:28
  Speaker 1: is out. Stuff you Should Know is a production of
• 11:34
  Speaker 1: I Heart Radio. For more podcasts my heart Radio, visit
• 11:37
  Speaker 1: the i heart Radio app, Apple Podcasts, or wherever you
• 11:40
  Speaker 1: listen to your favorite shows.