Short Stuff: Birthday Paradox

The Birthday Paradox involves math, so you know this one will go perfectly.

See omnystudio.com/listener for privacy information.

Episode Transcript

Speaker 1 (00:04):
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

(00:24):
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

(00:45):
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

(01:06):
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

(01:27):
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

(01:47):
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

(02:08):
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

(02:30):
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

(02:53):
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

(03:14):
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

(03:35):
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

(04:00):
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

(04:21):
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,

(04:50):
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

(05:12):
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,

(05:36):
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.

(05:56):
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.

(06:18):
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

(06:43):
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

(07:05):
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

(07:25):
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

(07:46):
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,

(08:07):
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

(08:28):
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

(08:50):
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.

(09:13):
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

(09:36):
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,

(09:59):
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,

(10:23):
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,

(10:45):
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

(11:07):
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

(11:28):
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.