Another classic puzzle if it's all right? This time I want your gut reaction and I encourage viewers
to note their gut reaction before you ponder further. It's one of those classic ones where you
should get gut reaction, park it, and then analyse whether it has any sense to it. And this time
I'm going to ask you a probability question. I'd like you to answer this question on the scale of
impossible (i.e. probability zero) up to certain; and I don't really care about a number. If you can give
me a number I'll be impressed. The 50/50 is there; let's call this very unlikely, unlikely, likely,
very likely - I think you get the idea. So that's your scale to use, and here comes the question:
What are the chances that there are two people in London with the same number of hairs on their head?
Remember you don't need to give me a number I want somewhere on this scale.
(Brady: Very close to zero.) - Okay so I'm gonna put a Brady BH like down here, is that- is that fair? Or do
you want to be closer to zero than that? - (No I- I'd maybe go even a shade closer but that'll do for now.)
We're talking ballparks only, but the idea of actually giving me a percentage answer seems
nonsense as well I think. So my claim is - which I will try and answer in the remainder of the
video - is that I know exactly where on the scale the answer is and I know to full precision,
full precision - like I can nail the absolute full decimal expansion if you wanted me to.
And I'm just checking the viewers have got your gut reactions; you may disagree with Brady, you may
agree, uh maybe you disagree by like a fraction of percent - who knows? So uh the reason I'm asking
this question is that there's a little bit of background I want to explain. In the UK there's
a qualification you can take post-16, if you're not doing an A-level in maths you can do something called
core maths now, it's a relatively recent thing. It's the matter of everyday life and it's really useful
for students to take who are not going to do an A-level in maths but still want to keep some basic
numeracy, understand headlines. It sounds like a sales pitch a little bit because I think more
people should do it but the point is part of that course is estimation skills. Like you get examined
how well you can estimate things and there's these classic types of questions called Fermi
questions or Fermi estimations where- the story is that Enrico Fermi, the physicist, estimated
this size of an atomic blast. He didn't have any other way of measuring it so he dropped a piece
of paper, noticed how far the blast wind blew it, and got the size of the blast within a factor of
2. Detail didn't matter but the size estimate was a really useful skill and actually knowing roughly
how many things occur; maybe I don't know like the number of people in London or maybe the hairs on
your head, those questions are tied up with this. So we're going to use a bit of Fermi estimation to
solve this once and for all. - (It wasn't a trick question, you're not counting people with zero)
(hairs on their head are you?) - Well if we did what's your answer? - (Well then then it's 1 because there)
(are two people with zero, but I'm not counting that.) - Brady, inexplicably that's at the other end of the
scale from your guess, but let's say to be fair to you that we are not counting that and maybe we should
rephrase the question. Most people are shouting at the screen probably I think "What about bald people?"
So if we said what are the chances of two hairy people - whatever that means - having the same
number of hair follicles on their head whatever we could have hair follicles- whatever. We're going to
do some estimation in typical core maths style. How many hairs are on our heads? Like are we talking
10 million? Are we talking 10? - (I'm regretting my decision now. There aren't that many hairs on a head)
Well so I had a thought about how to give people advice; estimating stuff is hard which is
incidentally why they use these sort of questions as interview questions quite often, like how many
tennis balls can you fit on a jumbo jet I've heard was a Google interview question. No
one cares about the answer they care about how you think about it. So maybe you come up with your own
ways of thinking about hairs on your head; I had an idea which is: I had a 30 centimeter ruler in
my drawer, a nice bendy one, and I can sort of like- well it's it's roughly- my hair is roughly like 30
centimeters by 30 centimeters. I mean I know my hair is a source of contention among Numberphile
viewers - there's been incidences on camera where I had questionable hair and possibly still
do - but we're leaving that aside. - (You think I haven't just put them on screen? That's cute.)
I talked myself into that didn't I? New Numberphile video, new haircut - that's my motto.
And I think actually if you drill down, the size of my head is kind of within grasp of centimeters and
if you think about like a one centimeter squared patch - no close-ups right - I think we can get an
order of magnitude. So I think there's probably more than 10 hairs in a centimeter squared, like
it like feels like sparse. Probably not as many as a thousand, so let's pick a hundred. I'm going
to start writing some stuff down. 100 hairs per centimeter squared, and then roughly let's get
the size of my head. So when- I've already waved the rule around, that's 30 centimeters by 30 centimeters
approximately 900 centimeters squared? Times 100 - 90,000 hairs. How do you feel about that?
(Loads of people in London have the same number of hairs on their head.) - Are you wanting to change
your answer? I'm glad we are already acknowledging that there is doubt creeping into your earlier gut
reaction. So let's just settle it once and for all; uh I don't even trust my estimation but if we went
to - let's call it 100,000 - I don't think many people are going to be in order of magnitude
out from that. Incidentally I did do some research on this and apparently redhead people have uh
fewer hairs? Quality not quantity apparently is the name. But roughly people are estimating between
50,00 and 150,000 so I'm really glad our estimate is pretty good.
How many people in London, Brady? - (What is it? Is it like 12 million or 8 million? - And anyone who was shouting the
screen earlier when I asked the question would be like "It depends what you define as London" right?
Greater London, the city of London - however you really do it I think we're in millions, agreed?
Like even if you go small - and let's say greater London - I reckon about 8 million?
But again I don't actually care because I'm just going to say 8 million. You will see that
the actual detail doesn't really matter. Now you've already upgraded your estimate,
but let's think about why. I'm going to do what they call in the trade a piece of common sense,
uh which is the name for a mathematical theorem which everyone thinks is really profound and
everyone else is like - well that's just obvious. But this is called the pigeonhole principle and if you
have heard of it uh well done, you might think it's common sense. If you haven't heard of it it's time
you heard of it. So if there are 9 pigeon holes in this little grid here and I had 8 pigeons,
they could arrange themselves all through and there'd be a gap, they've got one pigeonhole each.
They could arrange themselves to double up, there's no rules against it if they're particularly cozy,
but they don't have to. My point is if I have 9 pigeons they could have a pigeonhole each. They
might double up but they don't have to; whereas if I have 10 pigeons, 2 pigeons are sharing.
At least. This is known as common sense to most people, and other people call it the pigeonhole
principle, and it's a surprisingly profound useful result uh in very different situations. If you've
got more things than options then the people choosing options are going to double up. And I
think- by the velocity of your nodding you have seen how this might apply. Let's imagine all the
people in London are choosing an amount of hairs; even if they're making up a number, right, they
just- I have 963 hairs and I have 900,000 whatever. - (They want to be unique.) - They're trying to be but
if 8 million people try it and there's only roughly 100,000 or even 200,000
or even 300,000 options two of them are going to duplicate. There's nothing we can do about
it. I'm sorry to break it to you, not only we're not down here we are precisely there. It's 100% and
I said I'd do the full decimal expansion if you want it but it gets pretty dull pretty quickly.
So I like this question because it feels like our gut reaction goes wrong quickly and I think
it's because we're not very good at estimating without training - incidentally why core maths is
a good thing to study. And it's nice to actually do an exam where you're rewarded in this common
sense way of thinking. - (I studied maths Ben and I got it wrong terribly.) - I think- I think mathematicians
particularly- you've done your share of maths, maybe more than the majority of people in the world,
are actually less good at the basic common sense estimating than other people. And people who have
to do a lot of this every day are better at it, it just takes practice. And actually studying maths
doesn't involve lots of it which is why there's an interesting qualification to talk about and a good
skill to have whatever qualifications you study. The art of estimation is important and in this
case what's nice is the probability question not only has a definitive answer; all you need is rough
ballpark estimates to nail it down completely. It turns out that the pigeonhole principle has lots
of important applications if you want to look at the possibility of things duplicating, if you make
up a random password with a certain key space and someone else makes it - like what are the chances
of duplication occurring? Suddenly you realise it's the same sort of question and that's important. And
there are links to another cliche problem called the Birthday Paradox, like what are the chances two
people share a birthday? It's not quite the same but if you have enough people in that question, if
you have 366 people in a room, you guarantee that two of them share a birthday. Same idea, pigeonhole
principle means at least one is going to double up. That's it, that's the hairy problem for you
and I think it's a good advert for estimating skills and this qualification called Core Maths.
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Look at that, I could watch that all day.
Brady, what are the chances of 2 people in London having exactly the same number of hairs on their head?
(100% because of the pigeonhole principle.) - Well this video is really short.
There we go.