Normal distribution excel exercise

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Kind: captions
Language: en

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In this video we're going to
cover what is arguably, the

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single most important concept
in all the statistics.

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Well if you go into almost any
scientific field you might even

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argue that it's the single
most important concept.

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I've actually told people that
it's kind of sad they don't

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cover this in the
core curriculum.

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Everyone should know about is
because it touches on every

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single aspect of our lives and
that's the normal distribution

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or the Gaussian distribution
or the bell curve.

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And just to kind of give you a
preview of what it is, my

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preview will actually make it
seem pretty strange but as we

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go through this video hopefully
you'll get a little bit more

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intuition of what
it's all about.

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The Gaussian distribution or
the normal distribution,

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they're two words
for the same thing.

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It was actually Gauss
who came up with it.

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I think he was studying
astronomical phenomenon

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when he did.

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But it's a probability density
function just like we studies

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the Poisson distribution.

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It's just like that.

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And just to give you the
preview it looks like this.

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The probability of getting
any x, and it's a

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class of probability
distribution functions.

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Just like the binomial
distribution is and the Poisson

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distribution, it's based on
a bunch of parameters.

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This is how you would
traditionally see it written in

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a lot of textbooks and if we
have time, I'd like to

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rearrange the algebra just to
get a little bit more intuition

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of how it all works out.

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Or maybe we could get
some insights on where

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it all came from.

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I'm not going to prove it in
this video, that's a little

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bit beyond our scope.

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Although, I do want to do it
and there's actually some

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really neat mathematics
that might show up.

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If you're a math lead there's
something called Sterling's

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formula what you might want
to do a Wikipedia search on,

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which is really fascinating.

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It approximates factorials
with essentially a

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continuous function.

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But I won't go into
that right now.

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The normal distribution is 1
over -- this is how it's

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normally written -- the
standard deviation times the

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square root of 2 pi times
e to the minus 1/2.

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Well, I like to write it this
way, it easier to remember,

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times whatever value you're
trying to get minus the mean of

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our distribution divided by the
standard deviation of our

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distribution squared.

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And so if you think about it,
actually this is a good thing

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to just notice right now.

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This is how far I'm from the
mean and we're dividing that

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by the standard deviation of
whatever our distribution is.

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This is a preview of actually a
normal distribution that I've

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plotted, the purple line here
is a normal distribution.

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Initially the whole exercise --
I know I jump around a little

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bit -- is to show you that the
normal distribution is a good

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approximation for the binomial
distribution and vice versa.

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If you take enough trials in
your binomal distribution and

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we'll touch on that a second.

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The intuition of this term
right here I think is

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interesting because we're
saying, how far are we away

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from the mean, we're dividing
by the standard deviation.

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So this whole term right here
is how many standard deviations

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we are away from the mean.

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This is actually called
the standard z score.

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One thing I found in statistics
is there's a lot of words a lot

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of definitions and they all
sound very fancy, the

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standard z score.

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But the underlying concept
is pretty straightforward.

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Let's say I had a probability
distribution and I get some x

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value that's out here and it's
3 and a half the standard

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deviations is away from the
mean, then it's standard

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z score is 3 and a half.

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Anyway let's focus on the
purpose of this video.

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So that's what the normal
distribution, I guess

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the probability density
function for the normal

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distribution looks like.

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But how did it get there?

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By the end of this video you
should at least feel

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comfortable that this is a good
approximation for the binomial

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distribution if you're
taking enough trials.

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And that's what's fascinating
about the normal distribution

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is that if you have the sum --
and I'll do a whole other video

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on the central limit theorem --
but if you have the sum of many

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independent trials approaching
infinity, that the distribution

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of those, even though the
distribution of each of those

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trials might have been
non-normal but the distribution

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of the sum of all those trials
approaches the normal

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distribution.

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I'll talk more
about that later.

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But that's why it's such a good
distribution to kind of assume

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for a lot of underlying
phenomenon.

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If you're kind of modeling
weather patterns or drug

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interactions and you we'll talk
about where it might work well

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and where it might
not work so well.

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Like sometimes people might
assume things like a normal

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distribution in finance and
we've see the financial crisis

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that's led to a lot of
things blowing up but.

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Anyway, let's go back to this.

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This is a spreadsheet
right here.

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I just made a black background
and you can downloaded it at

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khanacademy.org/downloads
Actually, if you just do that

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you'll see all of
the downloads.

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I haven't put it there yet, I'm
going to do it right after I

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record the videos
downloads/normal

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distribution.xls.

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If you just go up to
khanacademy.org/download/

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you'll see all the things
there and you'll see

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this spreadsheet.

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I encourage you to play with it
and maybe do other spreadsheets

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were you experiment with it.

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So this spreadsheet what we do
is we're doing a game or let's

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say I'm sitting I'm on a street
and I flip a coin, I flip

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a completely fair coin.

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If I get heads, this is heads,
I take a step backwards or

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let's say a step to the left.

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And if I get a tails I
take a step to the right.

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So in general I always have a
-- this is a completely fair

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coin -- I have a 50 percent
chance of taking a step to the

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left and I have a 50 percent
chance of taking a

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step to the right.

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So your intuition there is if
I told you I took a you a

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thousand flips of the coin
you're going to keep

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going left and right.

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If by chance you get a bunch
of heads, you might end up

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really kind of moving
over to the left.

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If you get a bunch of tails you
might move over to the right.

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And we learned already the odds
of getting a bunch of tales or

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many more tails than heads is a
lot lower than things kind of

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being equal or close to equal.

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Right here what I've done --
let me scroll down a little

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bit, I don't want to lose the
whole thing -- is I have this

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little assumption here and I
encourage you to fill that out

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and change it as you like.

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This is the number
of steps I take.

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This is the mean number of left
steps and all I did is I got

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the probability and we
figured out the mean of the

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binomial distribution.

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The mean of the binomial
distribution is essentially the

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probability of taking a left
step times the total

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number of trials.

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So that's equal to 5, that's
where that number comes from.

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And then the variance -- and
I'm not sure if I went over

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this and I need prove this to
you if I have and I'll make a

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whole other video on the
variance of the binomial

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distribution -- is essentially
equal to the number of trials,

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10 times the probability of
taking the left step or kind of

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a successful trial -- I'm
defining left as a successful

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trial, that could be right as
well -- times the probability

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of 1 minus the successful trial
or non successful trial.

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In this case they're equally
probable and that's where

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I got the 2.5 from.

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And that's all on
the spreadsheet.

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If you actually click on
the cell and look at the

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actual formula I did that.

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Although sometimes when you
see it in Excel it's a

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little bit confusing.

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And this is just the square
root of that number.

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The standard deviation
is just the square

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root of the variance.

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That's just the
square root of 2.5.

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And so if you look here
this says, OK what is

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the probability that
I take 0 steps?

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So I take a total of 10 steps
-- just to understand this

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spreadsheet -- what is the
probability that I

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take 0 left steps?

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And just to make clear, if I
take 0 left steps that means I

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must have taken 10 right steps.

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And I calculate this
probability -- I should have

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drawn maybe a line here -- I
calculate this using the

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binomial distribution.

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And how do I do that?

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Let me actually switch
colors just to make

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things interesting.

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Do they have a purple here?

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I'll do a blue.

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So blue for binomial.

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So what I have here is
how many total steps?

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There's a total of 10 steps.

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So 10 factorial, that's kind of
the number of trials I have.

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Of that I'm choosing
0 to go left.

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So 0 factorial divided by
10 minus 0 factorial.

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This is 10 choose 0.

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I'm choosing 0 left steps of
the total 10 steps I'm taking

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times the probability of 0 left
steps so, it's the probability

00:09:23.900 --> 00:09:28.110
of a left step, I'm only taking
0 of them times the probability

00:09:28.110 --> 00:09:31.550
of a right step, and I'm
taking 10 of those.

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So that's where this number
came from, this .001.

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That's what the binomial
distribution tells us.

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And then this one similarly, is
10 factorial over 1 factorial

00:09:44.530 --> 00:09:47.070
over 10 minus 1 factorial.

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That's how I get that one.

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And once again, if you click
on the actual cell you'll

00:09:50.650 --> 00:09:51.960
see that explained.

00:09:51.960 --> 00:09:53.020
We've done this multiple times.

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This is just a
bionomial calculation.

00:09:54.490 --> 00:09:59.080
Then right here, after
this line right here, you

00:09:59.080 --> 00:10:00.520
can almost ignore it.

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I did that so that I can do a
bunch of different scenarios.

00:10:02.930 --> 00:10:09.490
For example, if I were to go to
my spreadsheet, and instead of

00:10:09.490 --> 00:10:17.580
doing 10 I wanted to do 20
steps then everything changes.

00:10:17.580 --> 00:10:23.410
And that's why down here after
you get to a certain point the

00:10:23.410 --> 00:10:26.130
whole thing just
kind of repeats.

00:10:26.130 --> 00:10:28.000
I'll let you think
about why I do that.

00:10:28.000 --> 00:10:30.200
Maybe I should have made
a cleaner spreadsheet.

00:10:30.200 --> 00:10:33.490
But it doesn't affect the
scatter plot chart that I did.

00:10:33.490 --> 00:10:38.170
And so this plot in blue, and
you can't see it because the

00:10:38.170 --> 00:10:39.590
purple is almost right over.

00:10:39.590 --> 00:10:43.560
Actually let me make it
smaller so that you can see.

00:10:43.560 --> 00:10:48.080
Let's say I only took 6 steps.

00:10:48.080 --> 00:10:51.200
Well it's still hard to see the
difference between the two.

00:10:51.200 --> 00:10:54.590
Once again the whole point
of this is to see that the

00:10:54.590 --> 00:10:56.570
normal distribution is
a good approximation.

00:10:56.570 --> 00:10:58.730
But they're so close that
you can't even see the

00:10:58.730 --> 00:10:59.430
difference on mine.

00:10:59.430 --> 00:11:01.705
If you only took four steps,
OK, I think you can see here.

00:11:04.710 --> 00:11:06.460
Let me get my screen draw on.

00:11:10.320 --> 00:11:12.830
The blue curve is
right around there.

00:11:12.830 --> 00:11:14.910
This is the binomial.

00:11:14.910 --> 00:11:16.910
There's only a few points
here, you the points

00:11:16.910 --> 00:11:18.560
only go up to here.

00:11:18.560 --> 00:11:21.890
This is if I take 0 steps left,
1 step left, 2 steps left,

00:11:21.890 --> 00:11:23.330
3 steps left, 4 steps left.

00:11:23.330 --> 00:11:25.750
And then I plot it and then I
say what's the probability

00:11:25.750 --> 00:11:27.840
using the binomial
distribution?

00:11:27.840 --> 00:11:29.710
And this is my final
position right?

00:11:29.710 --> 00:11:33.460
If I take 0 steps to the left
then I take 4 steps to the

00:11:33.460 --> 00:11:36.090
right so my final position
is at 4, so that's the

00:11:36.090 --> 00:11:37.950
scenario right here.

00:11:37.950 --> 00:11:40.220
Let me switch my color back to
yellow, it's easier to see.

00:11:44.490 --> 00:11:49.730
If I take 4 steps to the left,
I take 0 steps to the right and

00:11:49.730 --> 00:11:52.780
so my final position is
going to be at minus 4.

00:11:52.780 --> 00:11:54.030
It's going to be here.

00:11:54.030 --> 00:11:58.980
If I take an equal amount of
both, that's this scenario,

00:11:58.980 --> 00:12:00.500
then I'm neutral.

00:12:00.500 --> 00:12:02.840
I'm just stuck in the
middle right here.

00:12:02.840 --> 00:12:05.360
I take 2 steps to the right and
then I take 2 steps to the left

00:12:05.360 --> 00:12:07.630
or vice versa, I take 2 steps
to the left and then I take

00:12:07.630 --> 00:12:09.760
2 steps right and I
end up right there.

00:12:09.760 --> 00:12:12.720
Hopefully that makes
a little sense.

00:12:12.720 --> 00:12:13.790
My phone is ringing.

00:12:13.790 --> 00:12:17.060
I'll ignore that because
the normal distribution

00:12:17.060 --> 00:12:17.990
is so important.

00:12:17.990 --> 00:12:21.050
Actually, my 9 week old son is
watching so this is the first

00:12:21.050 --> 00:12:23.200
time I have a live audience.

00:12:23.200 --> 00:12:27.300
He might pick up something
about the normal distribution.

00:12:27.300 --> 00:12:30.650
So the blue line right here --
I'll trace it maybe in yellow

00:12:30.650 --> 00:12:35.470
so you can see it -- is the
plot of the binomial

00:12:35.470 --> 00:12:36.500
distribution.

00:12:36.500 --> 00:12:40.620
I connected the lines but you
see the binomial distribution

00:12:40.620 --> 00:12:42.520
look something more like this.

00:12:42.520 --> 00:12:47.310
This is the probability
of getting to minus 4.

00:12:47.310 --> 00:12:51.770
This is the probability
of going to minus 2.

00:12:51.770 --> 00:12:55.390
This right here is
the probability of

00:12:55.390 --> 00:12:57.240
ending up nowhere.

00:12:57.240 --> 00:13:06.030
Then this is the probability of
ending up 2 to the right and

00:13:06.030 --> 00:13:09.510
this is the probability of
ending up 4 to the right.

00:13:09.510 --> 00:13:11.670
This is the binomial
distribution, I just plotted

00:13:11.670 --> 00:13:13.860
these points right here.

00:13:13.860 --> 00:13:14.490
This is 0.375.

00:13:14.490 --> 00:13:16.520
This is 0.375.

00:13:16.520 --> 00:13:18.000
That's the height of that.

00:13:18.000 --> 00:13:21.320
Now what I wanted to show
you is that the normal

00:13:21.320 --> 00:13:24.530
distribution approximates
the binomial distribution.

00:13:24.530 --> 00:13:29.550
So this right here, I wanted
to say what does the normal

00:13:29.550 --> 00:13:34.880
distribution tell me is the
probability of ending up

00:13:34.880 --> 00:13:38.180
with exactly 0 left steps?

00:13:38.180 --> 00:13:42.260
This is a little bit tricky.

00:13:42.260 --> 00:13:44.970
The binomial distribution
is a discreet probability

00:13:44.970 --> 00:13:45.500
distribution.

00:13:45.500 --> 00:13:48.100
You can just look at this chart
or look here and you say, what

00:13:48.100 --> 00:13:56.280
is the probability of having
exactly 1 left step and 3 right

00:13:56.280 --> 00:13:58.160
steps which puts me right here?

00:13:58.160 --> 00:14:00.530
Well you just look at this
chart and you say oh, that

00:14:00.530 --> 00:14:02.300
puts me right there.

00:14:02.300 --> 00:14:05.100
I just read that probability,
it's actually .25.

00:14:05.100 --> 00:14:07.400
And I say oh, I have a 25
percent chance of ending

00:14:07.400 --> 00:14:12.010
up 2 steps to the right.

00:14:12.010 --> 00:14:14.090
There's a 25 percent chance.

00:14:14.090 --> 00:14:17.400
The normal distribution
function is a continuous

00:14:17.400 --> 00:14:20.180
probability distribution so
it's a continuous curve.

00:14:20.180 --> 00:14:22.130
It looks like that, it's a
bell curve and it goes off

00:14:22.130 --> 00:14:26.400
to infinity and starts
approaching 0 on both sides.

00:14:26.400 --> 00:14:28.090
It looks something like that.

00:14:28.090 --> 00:14:30.040
This is a continuous
probability distribution.

00:14:30.040 --> 00:14:31.800
You can't just take a point
here and say, what's the

00:14:31.800 --> 00:14:35.250
probability that I end
up 2 feet to the right?

00:14:35.250 --> 00:14:37.080
Because if you just say that
there's the actual the

00:14:37.080 --> 00:14:39.580
probability of being exactly --
and you should watch with my

00:14:39.580 --> 00:14:42.170
video on probability density
functions -- but the

00:14:42.170 --> 00:14:45.270
probability of being exactly 2
feet to the right, exactly, I

00:14:45.270 --> 00:14:48.500
mean I'm talking to the
atoms, is close to 0.

00:14:48.500 --> 00:14:51.430
You actually have to specify
a range around this.

00:14:51.430 --> 00:14:56.900
What I assume in this
within a half a foot

00:14:56.900 --> 00:14:58.330
in either direction.

00:14:58.330 --> 00:14:58.690
Right?

00:14:58.690 --> 00:15:00.380
If we're talking about feet.

00:15:00.380 --> 00:15:03.840
To figure that out what I did
here is I took the value of

00:15:03.840 --> 00:15:07.280
the probability density
function there.

00:15:07.280 --> 00:15:09.600
And I'll show you how
I evaluated that.

00:15:09.600 --> 00:15:12.450
And then I multiply that by 1.

00:15:12.450 --> 00:15:14.580
So it gives me this area.

00:15:14.580 --> 00:15:17.620
And I use that as an
approximation for this area.

00:15:17.620 --> 00:15:19.580
If you really want to be
particular about it what you

00:15:19.580 --> 00:15:23.000
would do is you would take the
integral of this curve between

00:15:23.000 --> 00:15:26.750
this point and this point
as a better approximation.

00:15:26.750 --> 00:15:27.970
We'll do that in the future.

00:15:27.970 --> 00:15:30.080
But right now I just want to
give you the intuition that the

00:15:30.080 --> 00:15:32.470
binomial distribution really
does converge to the

00:15:32.470 --> 00:15:33.380
normal distribution.

00:15:33.380 --> 00:15:36.290
So how did I get this
number right here?

00:15:36.290 --> 00:15:44.880
Well I said, what is the
probability that I think 1 left

00:15:44.880 --> 00:15:53.260
step -- I kind of used less
steps as success -- of one?

00:15:53.260 --> 00:15:59.080
And that equaled 1 over
the standard deviation.

00:15:59.080 --> 00:16:02.450
When I only took 4 steps the
standard deviation was 1.

00:16:02.450 --> 00:16:03.780
So 1 over 1.

00:16:03.780 --> 00:16:04.950
Actually let me change this.

00:16:08.330 --> 00:16:09.946
Let me change it to
a higher number.

00:16:13.840 --> 00:16:15.645
We'll go back to the
example where I'm at 10.

00:16:19.100 --> 00:16:20.552
So if this is at 10.

00:16:20.552 --> 00:16:22.560
Let me go back to
my drawing tool.

00:16:28.910 --> 00:16:30.980
Let me do this calculation.

00:16:30.980 --> 00:16:34.660
Actually, even better let
me do this calculation.

00:16:34.660 --> 00:16:37.160
So what's the probability
that I have 2 left steps?

00:16:37.160 --> 00:16:39.845
If I have 2 left steps I took a
total of 10 steps so I'm going

00:16:39.845 --> 00:16:42.800
to have 8 right steps and
that puts me 6 to the right.

00:16:42.800 --> 00:16:46.090
So that's this
point right here.

00:16:46.090 --> 00:16:47.170
So what's that probability?

00:16:47.170 --> 00:16:49.070
How do I figure this out
using the probability

00:16:49.070 --> 00:16:50.270
density function?

00:16:50.270 --> 00:16:51.150
How do I figure this height?

00:16:51.150 --> 00:16:56.150
Well I say the probability of
taking 2 left steps -- that's

00:16:56.150 --> 00:16:58.130
how I calculate it, if you
actually click on the cell

00:16:58.130 --> 00:17:04.150
you'll see that -- is equal to
1 over the standard deviation,

00:17:04.150 --> 00:17:11.360
1.581 -- and I just directly
reference the cell there --

00:17:11.360 --> 00:17:12.694
times the square root of 2 pi.

00:17:15.490 --> 00:17:18.885
I'm always in awe of the whole
notion of e to the i pi is

00:17:18.885 --> 00:17:19.940
equal to negative 1
and all of that.

00:17:19.940 --> 00:17:21.090
But there's another
amazing thing.

00:17:21.090 --> 00:17:25.530
That all of a sudden as we take
many trials we have this

00:17:25.530 --> 00:17:28.740
formula that has e and pi in it
and square roots but once again

00:17:28.740 --> 00:17:30.310
these two numbers just
keep showing up.

00:17:30.310 --> 00:17:33.290
It tells you something about
the order of the universe

00:17:33.290 --> 00:17:34.120
with a capital o.

00:17:34.120 --> 00:17:42.610
But let's see, times e to
the minus 1/2 times x.

00:17:42.610 --> 00:17:46.310
Well x is what we're trying
to calculate, two successes.

00:17:46.310 --> 00:17:54.780
To to have exactly 2 left,
so it's 2 minus the mean.

00:17:54.780 --> 00:18:01.000
So the mean is five, 2 minus
five divided by the standard

00:18:01.000 --> 00:18:10.160
deviation, divided by 1.581,
all of that squared.

00:18:10.160 --> 00:18:12.170
That's where this
calculation came from.

00:18:14.820 --> 00:18:18.590
So I told you in the last one
this right here just tells

00:18:18.590 --> 00:18:22.640
me this value up here.

00:18:22.640 --> 00:18:25.310
If I want to know this
exact probability,

00:18:25.310 --> 00:18:26.960
it's the area of this.

00:18:26.960 --> 00:18:29.010
And if I just take one
line the are is 0.

00:18:32.550 --> 00:18:36.010
Remember, in this case you can
only be 2 feet away because

00:18:36.010 --> 00:18:37.770
we're taking very exact steps.

00:18:37.770 --> 00:18:39.890
But what the normal
distribution is it's the

00:18:39.890 --> 00:18:44.280
continuous probability density
function so it can tell us

00:18:44.280 --> 00:18:48.720
what's the probability of
being 2.183 feet away?

00:18:48.720 --> 00:18:51.900
Which obviously can only happen
if we're taking infinitely

00:18:51.900 --> 00:18:53.330
small steps every time.

00:18:53.330 --> 00:18:54.420
But that's what it's use is.

00:18:54.420 --> 00:18:56.100
It happens when you start
taking an infinite

00:18:56.100 --> 00:18:56.940
number of steps.

00:18:56.940 --> 00:18:59.310
But it can approximate
the discreet.

00:18:59.310 --> 00:19:01.470
And the way I approximate it is
I say oh, what's probability of

00:19:01.470 --> 00:19:03.150
being within a foot of that.

00:19:03.150 --> 00:19:05.400
And so I multiply
this height, which I

00:19:05.400 --> 00:19:08.920
calculate here, times 1.

00:19:08.920 --> 00:19:13.380
So let's say this has a base of
1, to calculate this area which

00:19:13.380 --> 00:19:15.110
I use as an approximation.

00:19:15.110 --> 00:19:19.220
So you just multiply that times
1 and that's what you get here.

00:19:19.220 --> 00:19:20.240
And I just want to show you.

00:19:20.240 --> 00:19:26.050
Even with just 10 trials, the
curves, the normal distribution

00:19:26.050 --> 00:19:28.550
here is in purple and the
binomial distribution

00:19:28.550 --> 00:19:29.140
is in blue.

00:19:29.140 --> 00:19:31.070
So they're almost right
on top of each other.

00:19:35.610 --> 00:19:39.860
As you can take many more steps
they almost converge right on

00:19:39.860 --> 00:19:41.320
top of each other and I
encourage you to play

00:19:41.320 --> 00:19:42.700
with the spreadsheets.

00:19:42.700 --> 00:19:46.170
Actually, let me show
you that they converge.

00:19:46.170 --> 00:19:48.540
There's a convergence worksheet
on this spreadsheet as well if

00:19:48.540 --> 00:19:52.180
you click on the bottom
tab on convergence.

00:19:52.180 --> 00:19:54.450
This is the same thing but I
just want to show you what

00:19:54.450 --> 00:19:58.940
happens at any given point.

00:19:58.940 --> 00:20:02.550
Let me explain this
spreadsheet to you.

00:20:02.550 --> 00:20:04.040
So this is what's
the probability of

00:20:04.040 --> 00:20:06.570
moving left, right?

00:20:06.570 --> 00:20:09.060
So this is just saying, I'm
just fixing a point what's

00:20:09.060 --> 00:20:11.330
the probability -- and you
can change this -- of my

00:20:11.330 --> 00:20:13.120
final position being 10.

00:20:13.120 --> 00:20:18.720
And this essentially tells you
that if I take 10 moves, for my

00:20:18.720 --> 00:20:21.330
final position to be 10 to the
right, I have to take 10 right

00:20:21.330 --> 00:20:22.820
moves and 0 left moves.

00:20:22.820 --> 00:20:26.230
That's a typo right there, it
should be moves not movest.

00:20:26.230 --> 00:20:31.770
If I take 20 moves to end up 10
moves to the right then I have

00:20:31.770 --> 00:20:34.580
to make 15 right moves
and 5 left moves.

00:20:34.580 --> 00:20:37.780
Likewise if I take a total of
80 moves, if I think 80 flips

00:20:37.780 --> 00:20:41.580
of my coin to make me go left
or right, in order end up 10 to

00:20:41.580 --> 00:20:46.430
the right, I to take 45 right
moves and 35 left moves in any

00:20:46.430 --> 00:20:48.920
order and it will end up
with 10 to the right.

00:20:48.920 --> 00:20:52.920
So what I want to figure out
is, as I start taking a bunch

00:20:52.920 --> 00:20:58.650
of total moves -- here I max it
out at 170 -- if I start

00:20:58.650 --> 00:21:01.350
flipping this coin an infinite
number of times, I want to

00:21:01.350 --> 00:21:03.340
figure out what's the
probability that my final

00:21:03.340 --> 00:21:05.080
position is 10 to the right.

00:21:05.080 --> 00:21:08.880
And I want to show you that as
you take more and more moves

00:21:08.880 --> 00:21:11.730
the normal distribution becomes
a better and better

00:21:11.730 --> 00:21:15.260
approximation for the
binomial distribution.

00:21:15.260 --> 00:21:18.510
So right here, this calculates
the binomial probability, just

00:21:18.510 --> 00:21:21.040
the way we did before and you
can look at the cell

00:21:21.040 --> 00:21:21.770
to figure it out.

00:21:24.510 --> 00:21:26.170
I used left moves as a success.

00:21:30.120 --> 00:21:32.920
So this is 10 choose 0 and
we know what that is.

00:21:32.920 --> 00:21:36.840
It's 10 factorial over 0
factorial over 10 minus 0

00:21:36.840 --> 00:21:43.210
factorial times 0.5 to the
0 times 0.5 to the 10th.

00:21:43.210 --> 00:21:44.980
That's where that
number comes from.

00:21:44.980 --> 00:21:52.540
If I go to this one right
here is calculated.

00:21:52.540 --> 00:21:53.720
Actually let me write
it out because I think

00:21:53.720 --> 00:21:54.290
it's interesting.

00:21:54.290 --> 00:21:54.867
I have a total of 60 total
moves, so it's 60 factorial

00:21:54.867 --> 00:22:02.990
over, I have to have 25 left
moves so 25 factorial.

00:22:02.990 --> 00:22:09.510
So that's I'm 60 minus 25
factorial times the probability

00:22:09.510 --> 00:22:12.510
of a left move and have 25 of them, times the

00:22:12.510 --> 00:22:17.860
probability of a right move
and I have 35 of those.

00:22:17.860 --> 00:22:21.490
So that's just what the
binomial probability

00:22:21.490 --> 00:22:23.390
distribution will tell us.

00:22:23.390 --> 00:22:25.220
And then it figures out the
mean and the variance for each

00:22:25.220 --> 00:22:27.340
of those circumstances and you
could look at the formula.

00:22:27.340 --> 00:22:30.050
But the mean is just the
probability of having

00:22:30.050 --> 00:22:32.880
a left move times the
total number of moves.

00:22:32.880 --> 00:22:36.250
The variance is probability of
left times probability of right

00:22:36.250 --> 00:22:38.160
times total number of moves.

00:22:38.160 --> 00:22:40.670
And then the normal
probability, once again, I just

00:22:40.670 --> 00:22:43.810
use the normal probability.

00:22:43.810 --> 00:22:45.310
So I approximate
it the same way.

00:22:49.340 --> 00:22:51.620
And Excel has a normal
distribution function but I

00:22:51.620 --> 00:22:53.750
actually typed in the formula
because I wanted you kind of

00:22:53.750 --> 00:22:57.890
see what was under the covers
for that function that

00:22:57.890 --> 00:22:58.910
Excel actually has.

00:22:58.910 --> 00:23:04.770
So I actually say what's the
probability of 25 left moves?

00:23:04.770 --> 00:23:06.830
No, 45 left moves.

00:23:06.830 --> 00:23:14.690
So I say the probability of
45 left moves is equal to 1

00:23:14.690 --> 00:23:17.340
over the standard deviation.

00:23:17.340 --> 00:23:20.430
So in this situation the
standard deviation is

00:23:20.430 --> 00:23:21.720
the square root of 25.

00:23:21.720 --> 00:23:32.910
So it's five times 2 pi times e
to the minus 1/2 times 45 minus

00:23:32.910 --> 00:23:37.790
the mean, minus 50 over the
standard deviation, which we

00:23:37.790 --> 00:23:41.010
figured out was 5, squared.

00:23:41.010 --> 00:23:45.060
So that tells me the value of
what the normal distribution

00:23:45.060 --> 00:23:48.100
tells me for this situation
with this standard deviation

00:23:48.100 --> 00:23:51.020
and this mean and then I
multiply that by 1 -- you don't

00:23:51.020 --> 00:23:53.280
see that in the formula, I
don't actually write times 1 --

00:23:53.280 --> 00:23:55.020
to figure out the area
under the curve.

00:23:55.020 --> 00:23:57.380
Because remember it's a
continuous distribution

00:23:57.380 --> 00:23:58.480
function.

00:23:58.480 --> 00:24:01.840
This right here just gives me
the value but to figure out the

00:24:01.840 --> 00:24:03.590
probability of being within a
foot of it I have

00:24:03.590 --> 00:24:05.210
to multiply by 1.

00:24:05.210 --> 00:24:06.340
I'm approximating really.

00:24:06.340 --> 00:24:08.320
I really should take the
integral from there to there

00:24:08.320 --> 00:24:11.900
but this little rectangle is
a pretty good approximation.

00:24:11.900 --> 00:24:16.860
In this chart I show you that
as the total number of moves

00:24:16.860 --> 00:24:20.560
gets larger and larger the
difference between what the

00:24:20.560 --> 00:24:24.330
normal probability distribution
tells us and the binomial

00:24:24.330 --> 00:24:27.060
probability of distribution
tells us, gets smaller and

00:24:27.060 --> 00:24:30.930
smaller in terms of the
probability of you ending up

00:24:30.930 --> 00:24:32.100
to 10 moves to the right.

00:24:32.100 --> 00:24:35.040
And you can change
this number here.

00:24:35.040 --> 00:24:37.230
Let me change it
just to show you.

00:24:37.230 --> 00:24:38.370
You could say what's the
probability of being

00:24:38.370 --> 00:24:43.670
15 moves to the right?

00:25:02.680 --> 00:25:04.280
I think that something is
happening with the floating

00:25:04.280 --> 00:25:06.180
point error because when you
get to large factorials I

00:25:06.180 --> 00:25:08.770
think something weird
happens out here.

00:25:18.530 --> 00:25:20.600
You may just have to
get even further out.

00:25:20.600 --> 00:25:23.820
For 10 you can see clearly that
it converges and I'll trying to

00:25:23.820 --> 00:25:26.043
figure out why I was getting
those weird saw tooth patterns.

00:25:31.910 --> 00:25:35.110
Maybe while I do screen capture
something weird is happening.

00:25:35.110 --> 00:25:37.570
The whole point of this was to
show you that if you want to

00:25:37.570 --> 00:25:40.770
figure out the probability of
being 10 moves to the right, as

00:25:40.770 --> 00:25:44.840
you take more and more flips
of your coin the normal

00:25:44.840 --> 00:25:49.840
distribution becomes a much
better approximation for the

00:25:49.840 --> 00:25:51.730
actual binomial distribution.

00:25:51.730 --> 00:25:54.220
And as you approach infinity
they actually converge

00:25:54.220 --> 00:25:55.120
to each other.

00:25:55.120 --> 00:25:57.560
Anyway, that's all
for this video.

00:25:57.560 --> 00:25:59.290
I'll actually do several
more videos on the normal

00:25:59.290 --> 00:26:02.027
distributions because it is
such an important concept.

00:26:02.027 --> 00:26:02.380
See you soon.


Forehead numbers brain teaser

WEBVTT
Kind: captions
Language: en

00:00:01.730 --> 00:00:05.670
This is another brain teaser
that deals with people trying

00:00:05.670 --> 00:00:08.240
to logically deduce what's
on their forehead.

00:00:08.240 --> 00:00:15.939
And it's called the Forehead
Number Game.

00:00:19.960 --> 00:00:23.400
So what's going to happen is I
guess you could say the god of

00:00:23.400 --> 00:00:24.760
brain teasers is going
to get three

00:00:24.760 --> 00:00:25.790
people in a room together.

00:00:25.790 --> 00:00:28.250
Once again, these three people
are perfect logicians.

00:00:36.060 --> 00:00:38.710
Which means they have infallible
powers of logic.

00:00:38.710 --> 00:00:41.750
And they know that the other
two people-- they're three

00:00:41.750 --> 00:00:45.190
people in total --are also all
perfect logicians and have

00:00:45.190 --> 00:00:47.740
infallible powers of logic.

00:00:47.740 --> 00:00:48.910
We get them into the circle,
and we'll say

00:00:48.910 --> 00:00:49.790
this is the top view.

00:00:49.790 --> 00:00:52.630
So this is the top of
one guy's forehead.

00:00:52.630 --> 00:00:54.410
They're all bald for
convenience.

00:00:54.410 --> 00:00:56.910
To make it easy to paint
things on their head.

00:00:56.910 --> 00:00:58.780
This time we're not going
to paint blue

00:00:58.780 --> 00:01:00.700
paint on their forehead.

00:01:00.700 --> 00:01:02.790
We're actually going
to paint numbers.

00:01:02.790 --> 00:01:07.040
So you could say the dungeon
master of this game, or the

00:01:07.040 --> 00:01:10.510
the god of this game says I'm
going to pick three numbers.

00:01:10.510 --> 00:01:13.830
So three numbers
will be picked.

00:01:13.830 --> 00:01:15.560
They will be unique.

00:01:15.560 --> 00:01:19.260
Which means that no two of these
numbers are going to be

00:01:19.260 --> 00:01:19.880
the same number.

00:01:19.880 --> 00:01:22.030
There are going to be three
different numbers.

00:01:22.030 --> 00:01:25.210
They're all going to be
greater than zero.

00:01:25.210 --> 00:01:28.840
And this is key: One of the
numbers is going to be the sum

00:01:28.840 --> 00:01:30.090
of the other two.

00:01:38.950 --> 00:01:40.740
And they tell this to all
of the logicians.

00:01:40.740 --> 00:01:44.390
So this is the method that they
use to pick the number.

00:01:44.390 --> 00:01:46.940
And they tell this to the three
parties right there.

00:01:46.940 --> 00:01:48.790
And what they do is they take
these three numbers.

00:01:48.790 --> 00:01:50.970
And they write them on
people's foreheads.

00:01:50.970 --> 00:01:54.260
So say the numbers
are a, b and c.

00:01:54.260 --> 00:02:00.670
And we know that a plus-- let me
scroll down a little bit b

00:02:00.670 --> 00:02:03.030
is equal to c.

00:02:03.030 --> 00:02:05.640
We know that a, b and
c are different.

00:02:05.640 --> 00:02:08.490
And we know that a is
greater than zero.

00:02:08.490 --> 00:02:10.520
b is greater than zero.

00:02:10.520 --> 00:02:13.540
And c is also greater
than zero.

00:02:13.540 --> 00:02:15.160
What they do is they paint
them on their heads.

00:02:15.160 --> 00:02:17.790
So this guy will have
a on his forehead.

00:02:17.790 --> 00:02:19.820
This guy will have b
on his forehead.

00:02:19.820 --> 00:02:22.900
And let's say this guy has
c on his forehead.

00:02:22.900 --> 00:02:25.100
And so when this guy looks out--
and you know, the reason

00:02:25.100 --> 00:02:27.290
we're even putting things on
people's foreheads is because

00:02:27.290 --> 00:02:29.070
you can't see what's on
your own forehead.

00:02:29.070 --> 00:02:30.370
No tricks involved here.

00:02:30.370 --> 00:02:33.960
All you can see is what's on the
forehead of the other two

00:02:33.960 --> 00:02:35.100
people in the game.

00:02:35.100 --> 00:02:37.290
So this guy is going
to see a and b.

00:02:37.290 --> 00:02:40.470
This guy right here is going to
see the foreheads c and a.

00:02:40.470 --> 00:02:42.520
And then this guy's going
to see these two

00:02:42.520 --> 00:02:44.220
foreheads, c and b.

00:02:44.220 --> 00:02:45.690
So that's the rules
of the game.

00:02:45.690 --> 00:02:48.800
And of course, you don't know;
this guy, the number on his

00:02:48.800 --> 00:02:49.500
forehead is c.

00:02:49.500 --> 00:02:51.050
And it happens to be
the sum of a and b.

00:02:51.050 --> 00:02:51.960
But he doesn't know that.

00:02:51.960 --> 00:02:54.010
He doesn't know whether
it's the sum or not.

00:02:54.010 --> 00:02:56.950
He doesn't know he has
the sum number.

00:02:56.950 --> 00:02:58.580
This is just the way I
happened to draw it.

00:02:58.580 --> 00:02:59.740
So given those rules
of the game.

00:02:59.740 --> 00:03:01.415
And let's say you're
one of the players.

00:03:04.640 --> 00:03:08.110
And this is the view you see.

00:03:08.110 --> 00:03:10.200
You look at the two other
people in the game.

00:03:10.200 --> 00:03:14.020
Let me see if I can draw them
a little bit better.

00:03:14.020 --> 00:03:15.050
So you look at one person.

00:03:15.050 --> 00:03:15.740
This is your view.

00:03:15.740 --> 00:03:18.870
So this is one person
in the game.

00:03:18.870 --> 00:03:21.810
On his forehead-- which
is fairly large-- you

00:03:21.810 --> 00:03:28.430
see the number 20.

00:03:28.430 --> 00:03:34.630
And on the next guy, who
I'll draw in purple.

00:03:34.630 --> 00:03:43.640
The next guy, you see
the number 30.

00:03:43.640 --> 00:03:48.000
And what happens is-- so this
is literally your view.

00:03:48.000 --> 00:03:52.570
So they're seeing you with some
number on your forehead.

00:03:52.570 --> 00:03:54.400
And this guy can
see this 20 and

00:03:54.400 --> 00:03:55.370
whatever's on your forehead.

00:03:55.370 --> 00:03:58.210
This guy can see this 30 and
whatever's on your forehead.

00:03:58.210 --> 00:04:02.600
And the game show host, or the
dungeon master or soon.

00:04:02.600 --> 00:04:05.920
whoever, says, OK what's
on your forehead?

00:04:05.920 --> 00:04:08.970
And you look at these two
people, and you say, gee, I

00:04:08.970 --> 00:04:10.090
don't know what's
on my forehead.

00:04:10.090 --> 00:04:13.660
I'm a perfect logician, but I
still can't figure out what's

00:04:13.660 --> 00:04:15.450
on my forehead.

00:04:15.450 --> 00:04:17.930
And they say, OK fair enough.

00:04:17.930 --> 00:04:19.790
Then they go to this guy.

00:04:19.790 --> 00:04:23.010
We call him whatever we
want to call this guy.

00:04:23.010 --> 00:04:24.390
And they say, hey you,
do know what

00:04:24.390 --> 00:04:25.510
number's on your forehead?

00:04:25.510 --> 00:04:27.560
And he looks at whatever
number's on my

00:04:27.560 --> 00:04:28.350
forehead over here.

00:04:28.350 --> 00:04:29.500
Let's say this is my forehead.

00:04:29.500 --> 00:04:31.710
I don't know what number's
written on it.

00:04:31.710 --> 00:04:33.290
Some number's written up here.

00:04:33.290 --> 00:04:35.390
He looks at this number and then
at that guy, and says,

00:04:35.390 --> 00:04:38.620
you know what, I really don't
know what my number is.

00:04:38.620 --> 00:04:40.070
They say, OK fair enough.

00:04:40.070 --> 00:04:41.970
Then you turn to this guy.

00:04:41.970 --> 00:04:45.230
And you say, hey you, look
at this guy and that guy.

00:04:45.230 --> 00:04:46.450
Do you know what
your number is?

00:04:46.450 --> 00:04:48.340
And he looks at that number, and
he looks at the number on

00:04:48.340 --> 00:04:48.890
your forehead.

00:04:48.890 --> 00:04:49.620
And he says, you know what?

00:04:49.620 --> 00:04:51.615
I really don't know
what my number is.

00:04:51.615 --> 00:04:53.850
And then they come
back to you.

00:04:53.850 --> 00:04:58.060
And then you, as a perfect
logician says, oh, now I know

00:04:58.060 --> 00:04:59.980
what my number is.

00:04:59.980 --> 00:05:03.160
So my question to you-- and this
is the question of the

00:05:03.160 --> 00:05:05.570
brain teaser-- is, what
is your number?

00:05:05.570 --> 00:05:06.820
And why?

00:05:15.980 --> 00:05:17.110
So that's the question.

00:05:17.110 --> 00:05:20.130
And everything I'm going to do
from this point on in the

00:05:20.130 --> 00:05:22.750
video is essentially
going to be a hint.

00:05:22.750 --> 00:05:25.340
And then I'll actually
give the solution.

00:05:25.340 --> 00:05:26.410
So let's think about
it a little bit.

00:05:26.410 --> 00:05:27.760
And pause it, stop
it, whatever.

00:05:27.760 --> 00:05:28.730
Think about it for a while.

00:05:28.730 --> 00:05:30.620
And I would say this
is a fairly

00:05:30.620 --> 00:05:31.940
advanced level brain teaser.

00:05:31.940 --> 00:05:33.190
And it takes a lot of
trial and error.

00:05:33.190 --> 00:05:36.400
But the solution-- just as a
little bit of a hint-- isn't

00:05:36.400 --> 00:05:39.600
an ultre-complicated one, where
you have to know some

00:05:39.600 --> 00:05:43.160
higher level math.

00:05:43.160 --> 00:05:47.130
It's, once again,
very logical.

00:05:47.130 --> 00:05:49.010
You can make very logical
deductions here.

00:05:49.010 --> 00:05:51.100
You don't need any higher
math to really come

00:05:51.100 --> 00:05:53.550
to the correct answer.

00:05:53.550 --> 00:05:56.880
So let's lay it all out.

00:05:56.880 --> 00:05:59.970
So if I were to draw
the top view.

00:05:59.970 --> 00:06:02.360
We'll do it in yellow.

00:06:02.360 --> 00:06:03.890
This is you.

00:06:03.890 --> 00:06:07.420
This is the guy with
20 on his forehead.

00:06:07.420 --> 00:06:10.480
And this is the guy with
30 on his forehead.

00:06:10.480 --> 00:06:12.320
And what do we know?

00:06:12.320 --> 00:06:15.210
Let's call whatever's
on your forehead a.

00:06:15.210 --> 00:06:17.150
That's what you want
to figure out.

00:06:17.150 --> 00:06:21.260
So we know that a is
greater than zero.

00:06:21.260 --> 00:06:22.680
These are obviously
greater than zero.

00:06:22.680 --> 00:06:24.270
You see those already.

00:06:24.270 --> 00:06:27.110
We know that a can't
be 20 or 30.

00:06:27.110 --> 00:06:28.310
Now this is the interesting
thing.

00:06:28.310 --> 00:06:31.780
They told us that the numbers on
one of our foreheads is the

00:06:31.780 --> 00:06:35.400
sum of the numbers on the
other two foreheads.

00:06:35.400 --> 00:06:39.120
So either-- this
is a big hint.

00:06:39.120 --> 00:06:40.660
I think you might be able
to run with this after

00:06:40.660 --> 00:06:42.580
you get this hint.

00:06:42.580 --> 00:06:45.110
You say either a is a sum
of these two numbers.

00:06:45.110 --> 00:06:52.420
So a is equal to 20 plus
30, which would mean

00:06:52.420 --> 00:06:54.680
a is equal to 50.

00:06:54.680 --> 00:06:55.600
What's the other option?

00:06:55.600 --> 00:06:58.940
Either these two numbers, when
you sum them, is equal to a.

00:06:58.940 --> 00:07:00.230
That's one possibility.

00:07:00.230 --> 00:07:10.100
The other possibility is 20
plus a is equal to 30.

00:07:10.100 --> 00:07:12.500
Or, a is equal to 10.

00:07:12.500 --> 00:07:15.980
And of course I can't say that
30 plus a is equal to 20,

00:07:15.980 --> 00:07:18.800
because a can't be minus 10.

00:07:18.800 --> 00:07:20.150
That's the only other
possibility.

00:07:20.150 --> 00:07:21.930
And we already said that
all the numbers are

00:07:21.930 --> 00:07:22.550
greater than zero.

00:07:22.550 --> 00:07:26.840
So every one of these players
knows that their number on

00:07:26.840 --> 00:07:29.760
their forehead is either the sum
of the other two numbers,

00:07:29.760 --> 00:07:32.850
or it's the difference of
the other two numbers.

00:07:32.850 --> 00:07:34.250
This guy says, you know what?

00:07:34.250 --> 00:07:40.720
My number is either 30 plus a,
or it's the difference between

00:07:40.720 --> 00:07:42.510
a and 30, depending on
whichever is bigger.

00:07:42.510 --> 00:07:43.960
And he can see which
one's bigger.

00:07:43.960 --> 00:07:44.550
I can't see.

00:07:44.550 --> 00:07:48.900
So you can say the absolute
value of 30 minus a.

00:07:48.900 --> 00:07:50.090
He knows that about
his number.

00:07:50.090 --> 00:07:55.540
This guy, he knows that his
number is either 20 plus a, or

00:07:55.540 --> 00:07:58.695
the absolute value
of 20 minus a.

00:07:58.695 --> 00:08:03.730
And this all comes from the clue
or the rule that one of

00:08:03.730 --> 00:08:06.080
these numbers you is the sum
of the other two numbers.

00:08:06.080 --> 00:08:08.220
Although, you don't know if
you have the sum number on

00:08:08.220 --> 00:08:09.880
your forehead, or you don't know
if you have one of the

00:08:09.880 --> 00:08:12.030
numbers that make up the
sum on your forehead.

00:08:12.030 --> 00:08:14.860
But let's go back to our-- So
right from the beginning of

00:08:14.860 --> 00:08:18.000
the game, before anyone asks
us a question, we know that

00:08:18.000 --> 00:08:25.800
the number on our forehead
is either 50 or it is 10.

00:08:25.800 --> 00:08:29.370
And that's a pretty good hint,
and you might be able to take

00:08:29.370 --> 00:08:29.960
it from there.

00:08:29.960 --> 00:08:33.970
I mean, we've narrowed down the
universe of numbers from

00:08:33.970 --> 00:08:36.230
an infinite number of numbers
to two numbers.

00:08:36.230 --> 00:08:38.929
And this guy can just deduce
that automatically.

00:08:38.929 --> 00:08:42.360
So what other information
was given in the game?

00:08:42.360 --> 00:08:44.300
Well, we actually went
around the circle.

00:08:44.300 --> 00:08:46.670
So the first time you asked this
guy what number he had,

00:08:46.670 --> 00:08:47.740
he didn't know.

00:08:47.740 --> 00:08:49.990
He would be able to say, well
I either have 50 or 10, but

00:08:49.990 --> 00:08:52.740
that's not enough for me to make
a bold statement that my

00:08:52.740 --> 00:08:54.640
number is definitely 50,
or that my number

00:08:54.640 --> 00:08:55.360
is definitely 10.

00:08:55.360 --> 00:08:57.040
And so what other information
was there?

00:08:57.040 --> 00:08:59.730
We went to this guy, and
this guy wasn't able

00:08:59.730 --> 00:09:01.070
to guess his number.

00:09:01.070 --> 00:09:03.940
Then this guy wasn't able
to guess his number.

00:09:03.940 --> 00:09:06.730
And now the argument is that
now you should be able to

00:09:06.730 --> 00:09:07.160
guess your number.

00:09:07.160 --> 00:09:08.940
So let's think about
what happens.

00:09:08.940 --> 00:09:16.800
In the situation where this
guy is 50-- Well actually,

00:09:16.800 --> 00:09:17.860
let's take the other
situation.

00:09:17.860 --> 00:09:21.530
Let's take the situation
where this guy is 10.

00:09:21.530 --> 00:09:25.490
If this guy is 10, what
will this person see?

00:09:25.490 --> 00:09:29.050
And I'll do this situation
in green.

00:09:29.050 --> 00:09:32.280
This guy will say, I'm
either 30 plus 10.

00:09:32.280 --> 00:09:34.560
So he'll say, I'm either 40.

00:09:34.560 --> 00:09:36.230
Or I'm 30 minus 10.

00:09:36.230 --> 00:09:38.660
Or I'm 20, right?

00:09:38.660 --> 00:09:41.810
And he says, well, you know, no
real good conclusion I can

00:09:41.810 --> 00:09:42.170
make there.

00:09:42.170 --> 00:09:44.980
I could either be 40 or 20,
and I can't figure it out.

00:09:44.980 --> 00:09:46.130
Fair enough.

00:09:46.130 --> 00:09:47.630
Now you turn to this guy.

00:09:47.630 --> 00:09:51.030
In the situation where this
guy is 10, what's this guy

00:09:51.030 --> 00:09:52.160
going to say?

00:09:52.160 --> 00:09:53.730
He's going to say I'm
either 20 plus 10.

00:09:53.730 --> 00:09:58.880
I'm either 30, or I
am 20 minus 10.

00:09:58.880 --> 00:10:01.760
So 20 minus 10 would be 10.

00:10:01.760 --> 00:10:04.420
And you say, wait, wait, hold
on a second though.

00:10:04.420 --> 00:10:06.430
I can't be 10.

00:10:06.430 --> 00:10:09.920
Because if I were 10 and this
guy were 10, that would

00:10:09.920 --> 00:10:11.540
violate the rules of the game.

00:10:11.540 --> 00:10:13.990
One of the rules of the
game said that every

00:10:13.990 --> 00:10:15.480
number had to be unique.

00:10:15.480 --> 00:10:17.730
Every number had to
be different.

00:10:17.730 --> 00:10:23.290
So if you were 10-- you know
from the beginning to you're

00:10:23.290 --> 00:10:27.790
either 50 or 10-- if you are 10,
then when you go to this

00:10:27.790 --> 00:10:32.080
guy, he would be able to
deduce that he is 30.

00:10:32.080 --> 00:10:33.840
And how would he be able
to deduce that he's 30?

00:10:33.840 --> 00:10:38.540
Because he sees this
20 right here.

00:10:38.540 --> 00:10:41.000
If you were a 10, he'd
see a 20 and a 10.

00:10:41.000 --> 00:10:43.020
And he would say, I'm either
the sum of those numbers or

00:10:43.020 --> 00:10:44.080
the difference.

00:10:44.080 --> 00:10:45.860
If I'm the sum, I'm 30.

00:10:45.860 --> 00:10:47.690
If I'm the difference, I'm 10.

00:10:47.690 --> 00:10:51.480
But then he would say, I can't
be 10, because you're 10.

00:10:51.480 --> 00:10:53.030
And all the numbers
are different.

00:10:53.030 --> 00:10:54.960
So I have to be 30.

00:10:54.960 --> 00:10:55.570
Right?

00:10:55.570 --> 00:10:59.660
So in the reality where you are
10, when you go around the

00:10:59.660 --> 00:11:01.290
circle, this guy would
be able to figure out

00:11:01.290 --> 00:11:02.910
what number he is.

00:11:02.910 --> 00:11:03.200
Right?

00:11:03.200 --> 00:11:05.510
Because he's a perfect
logician.

00:11:05.510 --> 00:11:08.190
But in the problem statement I
gave you, we went around the

00:11:08.190 --> 00:11:10.790
circle and this guy couldn't
figure out what number he is.

00:11:10.790 --> 00:11:18.620
He couldn't establish the fact
that he is definitely 30.

00:11:18.620 --> 00:11:21.360
So given the fact that he
couldn't establish that he is

00:11:21.360 --> 00:11:25.160
30, it means that he did
not see a 10 here.

00:11:25.160 --> 00:11:26.580
He definitely did
not see a 10.

00:11:26.580 --> 00:11:29.740
If he saw a 10 there, he
would say, I am 30.

00:11:29.740 --> 00:11:31.600
Because I know I cannot be 10.

00:11:31.600 --> 00:11:34.180
So we know that you don't have
a 10 on your forehead, and so

00:11:34.180 --> 00:11:37.920
that's why when you come
back to you, you say, I

00:11:37.920 --> 00:11:39.900
know I must be 50.

00:11:39.900 --> 00:11:42.490
50 is the number on my forehead,
because if I were

00:11:42.490 --> 00:11:46.000
10, this guy would have been
able to get it before me.

00:11:46.000 --> 00:11:47.960
He would have been able to get
the number on his forehead

00:11:47.960 --> 00:11:49.900
before coming back to me.

00:11:49.900 --> 00:11:52.020
Anyway, I thought you would
enjoy that brain teaser.

00:11:56.530 --> 00:11:58.030
I didn't think of this one.

00:11:58.030 --> 00:12:01.300
I actually didn't think of any
of these, although they have a

00:12:01.300 --> 00:12:02.870
little bit of my twist
on some of them.

00:12:02.870 --> 00:12:05.760
But I'll be sure to add more.

00:12:05.760 --> 00:12:07.010
See


Alien abduction brain teaser

WEBVTT
Kind: captions
Language: en

00:00:01.850 --> 00:00:05.820
Ten of us have been abducted by
aliens because they view us

00:00:05.820 --> 00:00:08.500
as being representative
of the human race.

00:00:08.500 --> 00:00:10.690
And what they're going to do is
they're going to test us to

00:00:10.690 --> 00:00:14.910
see if the human race is
worthy of joining the

00:00:14.910 --> 00:00:18.280
intergalactic council
of planets.

00:00:18.280 --> 00:00:21.180
And if we're not, they're going
to kill us all and and

00:00:21.180 --> 00:00:24.090
turn Earth into a mushroom
farm, because

00:00:24.090 --> 00:00:25.720
that's their oil.

00:00:25.720 --> 00:00:28.200
So they take us together in
their spaceship and they say,

00:00:28.200 --> 00:00:31.870
you ten represent all of
humanity, and the fate of

00:00:31.870 --> 00:00:33.090
humanity rests in your hands.

00:00:33.090 --> 00:00:37.320
And we'll give you a test.
And the test is tomorrow.

00:00:37.320 --> 00:00:39.820
24 hours from right now, we're
going to put you all in a

00:00:39.820 --> 00:00:42.640
completely dark room,
and line you all up

00:00:42.640 --> 00:00:43.720
one behind the other.

00:00:43.720 --> 00:00:47.960
So if this is one guy-- let me
see if I can draw-- so if this

00:00:47.960 --> 00:00:49.370
is the guy in the front.

00:00:49.370 --> 00:00:54.070
He's looking a little worried.

00:00:54.070 --> 00:00:56.710
Someone described me as the
Bob Ross of mathematics.

00:00:56.710 --> 00:01:00.300
So I'll attempt to do a little
Bob Ross work right here.

00:01:00.300 --> 00:01:03.530
So this would be the
first of us.

00:01:03.530 --> 00:01:07.100
He has a business tie
on, and whatever.

00:01:07.100 --> 00:01:09.200
He's facing in this direction.

00:01:09.200 --> 00:01:10.460
And then there's ten
more of them.

00:01:10.460 --> 00:01:13.890
So instead of redrawing them,
let me just copy and paste it.

00:01:13.890 --> 00:01:17.580
We all look mysteriously
similar to each other.

00:01:17.580 --> 00:01:19.290
That's the second one.

00:01:19.290 --> 00:01:20.400
And the third one.

00:01:20.400 --> 00:01:21.690
There's ten of us like this.

00:01:21.690 --> 00:01:22.940
I'll do one more.

00:01:26.180 --> 00:01:27.160
Oh that's good enough.

00:01:27.160 --> 00:01:30.530
There's ten of us, and we're
essentially lined up front to

00:01:30.530 --> 00:01:33.560
back, so that this guy, all
he sees is the back

00:01:33.560 --> 00:01:34.360
of this guy's head.

00:01:34.360 --> 00:01:36.380
And he can also see the back
of this guy's head.

00:01:36.380 --> 00:01:39.530
Maybe there's a slight
incline, maybe the

00:01:39.530 --> 00:01:41.070
floor is like this.

00:01:41.070 --> 00:01:42.960
Let me see if I can draw it.

00:01:42.960 --> 00:01:44.880
The floor is at a slight
tilt, so one guy is

00:01:44.880 --> 00:01:46.370
standing right here.

00:01:46.370 --> 00:01:48.030
Another guy is standing
right here.

00:01:48.030 --> 00:01:49.170
Another guy's standing
right here.

00:01:49.170 --> 00:01:50.660
There's ten of us in a line.

00:01:50.660 --> 00:01:52.130
We're all facing in
that direction.

00:01:52.130 --> 00:01:56.050
All this guy right here
can see is the wall.

00:01:56.050 --> 00:01:57.780
This guy can see the two
guys in front of him.

00:01:57.780 --> 00:01:59.260
This guy can see the one
guy in front of him.

00:01:59.260 --> 00:01:59.920
There's ten of them.

00:01:59.920 --> 00:02:01.810
It goes all the way
back to ten

00:02:01.810 --> 00:02:02.830
So what they tell us
is we're going to

00:02:02.830 --> 00:02:03.950
line you up like this.

00:02:03.950 --> 00:02:06.900
They even draw a nice diagram
like this, because they want

00:02:06.900 --> 00:02:10.900
to give us a fair chance of
preventing Earth from turning

00:02:10.900 --> 00:02:12.080
into a mushroom farm.

00:02:12.080 --> 00:02:14.700
And they say, while the room
is completely pitch dark,

00:02:14.700 --> 00:02:18.040
we're going to place a hat
on each of your heads.

00:02:18.040 --> 00:02:24.960
And the hat will either
be mauve-- so this

00:02:24.960 --> 00:02:26.450
would be a mauve hat.

00:02:26.450 --> 00:02:31.260
Or they'll be green.

00:02:31.260 --> 00:02:35.970
So they either get a mauve
or a green hat on the

00:02:35.970 --> 00:02:36.720
back of your head.

00:02:36.720 --> 00:02:38.230
And the hat will be designed
in such a way, it

00:02:38.230 --> 00:02:39.210
won't have a big rim.

00:02:39.210 --> 00:02:41.800
So that you can't look
up and see your hat.

00:02:41.800 --> 00:02:45.000
This is alien technology, so it
is completely impossible to

00:02:45.000 --> 00:02:47.740
see the hat on your own head.

00:02:47.740 --> 00:02:50.040
And the distribution of the
hats will be random.

00:02:50.040 --> 00:02:51.570
It could be all green hats.

00:02:51.570 --> 00:02:54.690
It could be all-- well, I'll say
purple-- because I'm not

00:02:54.690 --> 00:02:57.720
quite sure how to
pronounce mauve.

00:02:57.720 --> 00:03:00.500
So it could be all purple hats,
it could be half purple,

00:03:00.500 --> 00:03:01.050
half green.

00:03:01.050 --> 00:03:02.900
It could be nine green
and one purple.

00:03:02.900 --> 00:03:03.410
They don't know.

00:03:03.410 --> 00:03:06.000
It could be any combination
of hats.

00:03:06.000 --> 00:03:07.760
And what they're going to do--
and they tell us all of this--

00:03:07.760 --> 00:03:10.520
is that they're going to start
with the person in the back.

00:03:10.520 --> 00:03:17.790
So this is the front wall,
this is the floor.

00:03:17.790 --> 00:03:20.960
This is the first guy right
here, second guy, third guy,

00:03:20.960 --> 00:03:25.400
fourth guy, fifth guy, sixth
guy, seven, eight, nine.

00:03:25.400 --> 00:03:26.270
So there's the tenth guy.

00:03:26.270 --> 00:03:28.200
They're all faced in
that direction.

00:03:28.200 --> 00:03:32.170
They're going to start with
this guy right here.

00:03:32.170 --> 00:03:35.460
And say, what color is
the hat on your head?

00:03:35.460 --> 00:03:43.550
And he'll answer, either
purple or green.

00:03:43.550 --> 00:03:44.790
He can only say purple
or green.

00:03:44.790 --> 00:03:46.770
I have to be exact here.

00:03:46.770 --> 00:03:49.720
He can only say purple
or green.

00:03:49.720 --> 00:03:53.120
And he can say it as
loud as he wants.

00:03:53.120 --> 00:03:56.940
But he has to say it at an even
tone, so that not more

00:03:56.940 --> 00:04:00.060
information can be conveyed,
other than the fact that he

00:04:00.060 --> 00:04:01.410
thinks his hat is
purple or green.

00:04:01.410 --> 00:04:03.780
Although this guy is close
enough that he can hear it.

00:04:03.780 --> 00:04:06.030
Everyone's going to hear what
everyone's going to say.

00:04:06.030 --> 00:04:07.540
But there's no intonation.

00:04:07.540 --> 00:04:10.330
And if you're right, if you say,
my hat is purple and you

00:04:10.330 --> 00:04:12.750
are wearing a purple
hat, you will live.

00:04:12.750 --> 00:04:15.410
And the 600 million
people that you

00:04:15.410 --> 00:04:17.339
represent will also live.

00:04:17.339 --> 00:04:18.709
And where they live
will not be turned

00:04:18.709 --> 00:04:20.339
into a mushroom farm.

00:04:20.339 --> 00:04:26.560
And if you're wrong, then they
will immediately-- what is

00:04:26.560 --> 00:04:27.720
that Star Trek weapon?

00:04:27.720 --> 00:04:28.610
Displace you.

00:04:28.610 --> 00:04:33.920
And they'll also do it to all
the people you represent.

00:04:33.920 --> 00:04:35.970
And turn all of their land
into mushroom farms. So

00:04:35.970 --> 00:04:39.140
there's this huge incentive
to try to get this right.

00:04:39.140 --> 00:04:41.240
So they say, we're going
to do this tomorrow.

00:04:41.240 --> 00:04:44.940
And your goal, obviously, is to
try to save as many lives

00:04:44.940 --> 00:04:45.940
as you can.

00:04:45.940 --> 00:04:48.390
And so we're going to put you
in a room, and you guys can

00:04:48.390 --> 00:04:50.880
discuss this for the next 24
hours, to come up with a

00:04:50.880 --> 00:04:55.470
system so that when we put you
in the dark room and we put

00:04:55.470 --> 00:04:57.265
the hats on each
of your heads.

00:05:00.220 --> 00:05:02.890
I shouldn't be drawing
brown hats.

00:05:02.890 --> 00:05:04.730
We'll put the hats
on your head.

00:05:04.730 --> 00:05:05.670
And then turn on the lights.

00:05:05.670 --> 00:05:06.860
I forgot to tell you that.

00:05:06.860 --> 00:05:09.430
They're going to put us in a
dark room, you can't see

00:05:09.430 --> 00:05:10.970
anything, we're all lined up.

00:05:10.970 --> 00:05:12.570
Then they're going to
turn on the lights.

00:05:12.570 --> 00:05:14.640
And so then they'll ask this
guy, what color is

00:05:14.640 --> 00:05:15.300
the hat on your head?

00:05:15.300 --> 00:05:17.240
He obviously can't look at his
own head, but he'll see

00:05:17.240 --> 00:05:19.030
everyone else's hats.

00:05:19.030 --> 00:05:22.890
And likewise this guy, he can't
see his own hat, and he

00:05:22.890 --> 00:05:24.320
can't see the guy behind him.

00:05:24.320 --> 00:05:26.720
But he'll hear what this guy
says, and he can see all of

00:05:26.720 --> 00:05:28.740
the hats in front of him.

00:05:28.740 --> 00:05:31.015
So they'll turn on the lights,
and then after this guy-- if

00:05:31.015 --> 00:05:33.700
he's right he lives, if he's
wrong he'll be displaced.

00:05:33.700 --> 00:05:36.170
And then they'll go
to the next guy.

00:05:36.170 --> 00:05:36.950
And then ask him.

00:05:36.950 --> 00:05:39.300
If he's right he lives,
and all the people

00:05:39.300 --> 00:05:40.230
he represents lives.

00:05:40.230 --> 00:05:42.800
If he's wrong he also
will get displaced.

00:05:42.800 --> 00:05:46.150
And they'll keep going all the
way up to the first guy.

00:05:46.150 --> 00:05:50.310
So our goal to essentially save
civilization, or at least

00:05:50.310 --> 00:05:54.750
human civilization, is to come
up with a system to save as

00:05:54.750 --> 00:05:57.900
many of us as possible.

00:05:57.900 --> 00:06:00.340
So that's the first take
of the brain teaser.

00:06:00.340 --> 00:06:03.690
So if you want to kind of
pause it right here.

00:06:03.690 --> 00:06:07.790
And actually before you pause
it, let me just clarify.

00:06:07.790 --> 00:06:10.650
All you can say is either
purple or green.

00:06:10.650 --> 00:06:11.600
No intonations.

00:06:11.600 --> 00:06:15.060
You can't say "pur-puuul" or
"purp-l" to kind of convey any

00:06:15.060 --> 00:06:16.220
type of information.

00:06:16.220 --> 00:06:18.830
All you can say is
purple or green.

00:06:18.830 --> 00:06:20.890
Or pronounce it in
some weird way.

00:06:20.890 --> 00:06:24.780
And there's no weird trick where
you can kind of look

00:06:24.780 --> 00:06:27.770
with the side of your eyes
or anything like this.

00:06:27.770 --> 00:06:31.850
You have to do it completely
based on what you hear and

00:06:31.850 --> 00:06:32.750
what you say.

00:06:32.750 --> 00:06:36.260
And if the aliens catch
any hint of you

00:06:36.260 --> 00:06:37.570
guys trying to cheat.

00:06:37.570 --> 00:06:38.350
Or all of us.

00:06:38.350 --> 00:06:39.570
I'm one of us.

00:06:39.570 --> 00:06:41.690
To cheat this, they'll just
evaporate everything and turn

00:06:41.690 --> 00:06:43.950
everything into mushroom
farms.

00:06:43.950 --> 00:06:45.580
So that's the problem
statement.

00:06:45.580 --> 00:06:48.940
How do you save as many
people as possible?

00:06:48.940 --> 00:06:51.800
And if you want to pause now,
feel free to do so.

00:06:51.800 --> 00:06:53.850
And now I'll give you kind of
a small hint, because this

00:06:53.850 --> 00:06:55.100
doesn't help you.

00:06:55.100 --> 00:06:58.070
It's easy to think of a lot of
solutions that might save a

00:06:58.070 --> 00:06:59.580
couple of people.

00:06:59.580 --> 00:07:03.370
But there's actually a solution
that will guarantee

00:07:03.370 --> 00:07:05.410
saving nine people.

00:07:05.410 --> 00:07:08.160
So that's the solution that
you have to come up with.

00:07:08.160 --> 00:07:11.490
And for the rest of this video
I'm just going to keep giving

00:07:11.490 --> 00:07:14.140
hints, and then eventually
give you the solution.

00:07:14.140 --> 00:07:18.770
So the goal is actually really
to save nine people.

00:07:18.770 --> 00:07:21.840
And you can which nine people
it's going to be.

00:07:21.840 --> 00:07:24.090
And so once again, pause it if
you don't want any more hints.

00:07:24.090 --> 00:07:26.630
It's going to be these
nine people.

00:07:26.630 --> 00:07:29.570
These nine people are guaranteed
to be saved.

00:07:29.570 --> 00:07:31.720
And this guy obviously is just
going to be guessing.

00:07:31.720 --> 00:07:34.590
He is the first guy to be asked
what color his hat is.

00:07:34.590 --> 00:07:35.790
He has no information.

00:07:35.790 --> 00:07:37.620
He can see the people in front
of him, but he has absolutely

00:07:37.620 --> 00:07:40.390
no information as to what
color this hat is.

00:07:40.390 --> 00:07:41.310
So he just guesses.

00:07:41.310 --> 00:07:45.520
So there's a 50% chance
that he's right.

00:07:45.520 --> 00:07:47.700
Without any other information,
we assume that there's a 50-50

00:07:47.700 --> 00:07:51.970
chance that the hat on the
top of your head is a

00:07:51.970 --> 00:07:53.480
purple or a green one.

00:07:53.480 --> 00:07:55.650
So your expected number of
people you're going to save is

00:07:55.650 --> 00:07:57.470
nine and a half.

00:07:57.470 --> 00:07:58.600
With the system.

00:07:58.600 --> 00:08:05.400
So these people are guaranteed
to be saved.

00:08:05.400 --> 00:08:07.100
And so, there's a couple
of things that are more

00:08:07.100 --> 00:08:08.760
probabilistic that don't
quite get you there.

00:08:08.760 --> 00:08:10.630
That might come to
mind immediately.

00:08:10.630 --> 00:08:12.140
And once again, stop
this video.

00:08:12.140 --> 00:08:14.660
Because it's fun to think
of these yourself.

00:08:14.660 --> 00:08:22.180
But this guy might just see,
OK how many green hats are

00:08:22.180 --> 00:08:24.630
there, and how many purple
hats are there?

00:08:24.630 --> 00:08:27.500
And then whichever there are
more of, he might say, oh my

00:08:27.500 --> 00:08:31.960
hat is-- let's say there's six
purple and three green in

00:08:31.960 --> 00:08:34.740
front of him-- he might
say my hat is purple.

00:08:34.740 --> 00:08:36.620
And he's either right
or wrong, and

00:08:36.620 --> 00:08:37.740
he might get displaced.

00:08:37.740 --> 00:08:40.700
But at least he'll have conveyed
that information to

00:08:40.700 --> 00:08:41.480
the rest of the group.

00:08:41.480 --> 00:08:44.159
That hey, the majority of the
hats are purple, so you're

00:08:44.159 --> 00:08:47.030
better off guessing purple than
green, and it'll save at

00:08:47.030 --> 00:08:48.330
least six of these
people because

00:08:48.330 --> 00:08:49.630
they'll all guess purple.

00:08:49.630 --> 00:08:53.000
And that's OK, and in this
situation you're guaranteed to

00:08:53.000 --> 00:08:55.320
save more than half of these
nine people, so you're

00:08:55.320 --> 00:08:58.490
guaranteed to save at least
five of these nine people.

00:08:58.490 --> 00:08:59.880
And then probablistically
maybe you can get

00:08:59.880 --> 00:09:03.360
up to seven or eight.

00:09:03.360 --> 00:09:04.620
Then there's the other method.

00:09:07.800 --> 00:09:09.055
Well, there's a ton of them.

00:09:09.055 --> 00:09:10.040
I don't go into them because
I realize I don't

00:09:10.040 --> 00:09:10.860
have a lot of time.

00:09:10.860 --> 00:09:14.820
So here is the real
deal solution.

00:09:14.820 --> 00:09:17.900
So really pause it and wait and
watch it tomorrow, unless

00:09:17.900 --> 00:09:19.390
you don't want the solution.

00:09:19.390 --> 00:09:24.510
The guy in the back,
what he does.

00:09:24.510 --> 00:09:26.670
So there are a bunch
of people.

00:09:26.670 --> 00:09:31.610
One, two, three, four,
five, six, seven,

00:09:31.610 --> 00:09:34.220
eight, nine, ten people.

00:09:34.220 --> 00:09:36.780
And they're all wearing hats.

00:09:36.780 --> 00:09:39.590
And remember, we have 24 hours
to come up with a system.

00:09:39.590 --> 00:09:40.870
So everyone has the system.

00:09:40.870 --> 00:09:43.100
And we don't know who's going
to be in what order also.

00:09:43.100 --> 00:09:43.770
That's the other thing.

00:09:43.770 --> 00:09:47.500
But that actually doesn't help
or hurt you much anyway.

00:09:47.500 --> 00:09:52.320
So the guy in the back, what
he does is he counts the

00:09:52.320 --> 00:09:58.280
number of green or
purple hats.

00:09:58.280 --> 00:09:59.770
Right?

00:09:59.770 --> 00:10:01.560
This is the system we've
come up with.

00:10:01.560 --> 00:10:04.300
And so this is true of whoever's
in the back.

00:10:04.300 --> 00:10:07.960
Whoever's in the back has
to do this, right?

00:10:07.960 --> 00:10:13.810
So if he counts that the number
of green-- let's say

00:10:13.810 --> 00:10:15.460
this is our system.

00:10:15.460 --> 00:10:18.080
He counts the green
actually only.

00:10:18.080 --> 00:10:20.270
So count the green.

00:10:20.270 --> 00:10:22.050
And he determines that
the number of

00:10:22.050 --> 00:10:26.650
green is an odd number.

00:10:26.650 --> 00:10:29.840
If he determines that the number
of green that he sees

00:10:29.840 --> 00:10:33.600
in front of him is an odd
number, then he is

00:10:33.600 --> 00:10:36.065
going to say green.

00:10:38.970 --> 00:10:41.690
For the guess on the
top of his head.

00:10:41.690 --> 00:10:49.190
If the number of green is
an even number, he's

00:10:49.190 --> 00:10:52.000
going to say purple.

00:10:52.000 --> 00:10:55.500
And of course, he doesn't know
what his hat color is, but at

00:10:55.500 --> 00:10:58.190
least, well, one,
he has a 50-50

00:10:58.190 --> 00:10:59.530
chance of saving himself.

00:10:59.530 --> 00:11:02.100
But now, what happens?

00:11:02.100 --> 00:11:05.260
Now, by giving this information
to the rest of the

00:11:05.260 --> 00:11:07.520
group, by screaming
green or purple.

00:11:07.520 --> 00:11:12.790
If this guy screams,
my hat is green!

00:11:12.790 --> 00:11:15.690
Then this guy, who's the next
guy in line, the next guy to

00:11:15.690 --> 00:11:20.370
be quizzed, knows that the
person behind him saw an odd

00:11:20.370 --> 00:11:21.805
number of green hats.

00:11:25.580 --> 00:11:27.890
Now this is interesting.

00:11:27.890 --> 00:11:32.030
If he counts the number of green
in front of him, and he

00:11:32.030 --> 00:11:37.160
also sees an odd number of green
hats, then that means

00:11:37.160 --> 00:11:41.020
that he was not one of the green
hats that this guy saw.

00:11:41.020 --> 00:11:45.210
So he must be a purple hat.

00:11:45.210 --> 00:11:47.840
So he'll just say, I
have a purple hat.

00:11:47.840 --> 00:11:49.390
And he will live.

00:11:49.390 --> 00:11:50.050
Right?

00:11:50.050 --> 00:11:53.210
And then you'll go
to this guy.

00:11:53.210 --> 00:11:55.300
And let's go the other
situation.

00:11:55.300 --> 00:11:58.560
Let's say that this guy,
he screamed green.

00:11:58.560 --> 00:12:01.180
Which means that he saw
an odd number of green

00:12:01.180 --> 00:12:02.000
hats in front of him.

00:12:02.000 --> 00:12:05.160
If this guy counts the green
hats of the eight people in

00:12:05.160 --> 00:12:08.010
front of him, and he says wow,
there's only an even number of

00:12:08.010 --> 00:12:12.000
green hats, he could say, wow
I must have a green hat.

00:12:12.000 --> 00:12:13.950
And so then he'll say green.

00:12:13.950 --> 00:12:16.200
And then he'll be right.

00:12:16.200 --> 00:12:22.645
Likewise, this guy-- let's say
it's the first situation that

00:12:22.645 --> 00:12:25.030
I said, this guy has
a purple hat on.

00:12:25.030 --> 00:12:27.810
So this guy saw an odd
number of green hats.

00:12:27.810 --> 00:12:29.640
He screamed green.

00:12:29.640 --> 00:12:33.010
And so this guy, he also sees
an odd number of green hats,

00:12:33.010 --> 00:12:35.000
because he's not one
of the green hats.

00:12:35.000 --> 00:12:37.190
So he says purple, he lives.

00:12:37.190 --> 00:12:44.290
This guy all of a sudden,
because this guy said purple.

00:12:44.290 --> 00:12:45.140
And he lived.

00:12:45.140 --> 00:12:46.760
I mean, you didn't
hear any buzzer.

00:12:46.760 --> 00:12:48.360
Actually you know he's going
to live, because

00:12:48.360 --> 00:12:50.820
the system will work.

00:12:50.820 --> 00:12:56.080
He knows that there must be an
odd number of green hats from

00:12:56.080 --> 00:12:58.410
him down forward, right?

00:12:58.410 --> 00:13:00.700
Because this guy didn't
say green.

00:13:00.700 --> 00:13:03.190
So if he knows that there's an
odd number of green hats from

00:13:03.190 --> 00:13:07.770
him down forward, and if he sees
an even number of green

00:13:07.770 --> 00:13:11.800
hats, he knows that he must be
one of the green hats, so

00:13:11.800 --> 00:13:13.250
he'll say green.

00:13:13.250 --> 00:13:18.030
And in that situation, if this
guy screamed green then that

00:13:18.030 --> 00:13:23.460
means that if this guy saw an
odd number of green hats from

00:13:23.460 --> 00:13:28.860
here on down, and one of these
guys behind him, or every time

00:13:28.860 --> 00:13:33.010
someone behind you says green
after that, then the number of

00:13:33.010 --> 00:13:40.270
green hats that are odd or even
in front of that person

00:13:40.270 --> 00:13:41.250
will change.

00:13:41.250 --> 00:13:44.410
And so they have to keep track
of that in their minds to

00:13:44.410 --> 00:13:46.820
figure out if the hats in front
of them are going to be

00:13:46.820 --> 00:13:47.430
green or purple.

00:13:47.430 --> 00:13:48.900
Actually let me give an
example because that's

00:13:48.900 --> 00:13:49.840
probably the easiest way.

00:13:49.840 --> 00:13:56.120
So there's one, two-- Actually
let me just do it in a neutral

00:13:56.120 --> 00:13:57.200
color so I don't waste time.

00:13:57.200 --> 00:14:01.770
So let's say the hats are green,
purple, purple, green,

00:14:01.770 --> 00:14:05.020
green, purple, green, green.

00:14:05.020 --> 00:14:05.440
How many is that?

00:14:05.440 --> 00:14:12.540
One, two, three, four, five,
six, seven, eight, nine, ten.

00:14:12.540 --> 00:14:16.280
So you go to the first
guy right here.

00:14:16.280 --> 00:14:17.210
And so he counts.

00:14:17.210 --> 00:14:18.660
He says, OK, how many
greens are there?

00:14:18.660 --> 00:14:23.270
There's one, two, three,
four, five.

00:14:23.270 --> 00:14:24.950
There are five greens.

00:14:24.950 --> 00:14:28.590
So the greens are odd.

00:14:28.590 --> 00:14:31.830
Which says that he is
going to say green.

00:14:31.830 --> 00:14:33.000
That's the system we've
come up with.

00:14:33.000 --> 00:14:35.650
So when they go up to this
guy, he says green.

00:14:35.650 --> 00:14:36.510
And he's wrong.

00:14:36.510 --> 00:14:39.530
So he actually gets
eliminated.

00:14:39.530 --> 00:14:41.870
And maybe the aliens were
willing to give us a pass on

00:14:41.870 --> 00:14:44.070
the first guy, but let's
say he gets eliminated.

00:14:44.070 --> 00:14:48.750
But he said green, so then this
guy on down knows that in

00:14:48.750 --> 00:14:50.960
front of that guy there was an
odd number of green hats.

00:14:50.960 --> 00:14:54.680
Now, this guy, he looks forward
and he counts one,

00:14:54.680 --> 00:14:59.800
two, three, four,
five green hats.

00:14:59.800 --> 00:15:03.040
So he also sees an odd number
of green hats, so he knows

00:15:03.040 --> 00:15:06.070
that he must not have a green
hat, because if he had a green

00:15:06.070 --> 00:15:08.750
hat, then it would turn the
number of green hats that he

00:15:08.750 --> 00:15:09.760
saw into even.

00:15:09.760 --> 00:15:11.360
So he doesn't have
a green hat.

00:15:11.360 --> 00:15:13.400
He says, I have a purple hat.

00:15:13.400 --> 00:15:14.650
And he lives.

00:15:17.520 --> 00:15:20.600
Now this guy, he counts
the number of green

00:15:20.600 --> 00:15:21.470
hats in front of him.

00:15:21.470 --> 00:15:25.230
One, two, three, four
green hats.

00:15:25.230 --> 00:15:28.910
He says, well gee, I only see an
even number of green hats.

00:15:28.910 --> 00:15:32.850
Up until this guy, he saw an
odd number of green hats.

00:15:32.850 --> 00:15:34.360
I must be a green hat.

00:15:34.360 --> 00:15:36.150
So he says green.

00:15:36.150 --> 00:15:37.740
Check.

00:15:37.740 --> 00:15:40.030
Now this guy's turn.

00:15:40.030 --> 00:15:44.400
So first of all, he knows how
many green hats this guy saw.

00:15:44.400 --> 00:15:48.240
He knows that this guy saw
exactly four green hats.

00:15:48.240 --> 00:15:50.950
Because if this guy had seen
five green hats, he would have

00:15:50.950 --> 00:15:51.910
said purple.

00:15:51.910 --> 00:15:54.810
So this guy knows that this
guy saw four green hats.

00:15:54.810 --> 00:15:58.590
But he sees only one, two,
three green hats.

00:15:58.590 --> 00:15:59.220
Right?

00:15:59.220 --> 00:16:03.160
So he knows that he must be
one of the green hats.

00:16:03.160 --> 00:16:04.710
So he screams green.

00:16:04.710 --> 00:16:06.530
So check.

00:16:06.530 --> 00:16:09.770
So then this guy knows, wow
there must be only three green

00:16:09.770 --> 00:16:11.560
hats from him on down.

00:16:11.560 --> 00:16:13.430
Because this guy must have
seen three green hats.

00:16:13.430 --> 00:16:15.480
If he saw four, he would
have said purple.

00:16:15.480 --> 00:16:19.050
So that guy uses that same logic
to say OK, there must be

00:16:19.050 --> 00:16:21.140
three green hats from
me on down.

00:16:21.140 --> 00:16:24.080
I see three green hats, so I
must not have one of them.

00:16:24.080 --> 00:16:25.220
I must be purple.

00:16:25.220 --> 00:16:27.820
And you just keep on going
down all the way until

00:16:27.820 --> 00:16:29.110
everyone answers correctly.

00:16:29.110 --> 00:16:31.850
And you save at least
nine people.

00:16:31.850 --> 00:16:33.340
And the bulk of humanity.

00:16:33.340 --> 00:16:35.890
And if you gave a pass
on the first guy,

00:16:35.890 --> 00:16:37.880
you save all of humanity.

00:16:37.880 --> 00:16:40.520
Anyway, I hope you found
that one enjoyable.

00:16:40.520 --> 00:16:42.760
This is one of my favorite
brain teasers.

00:16:42.760 --> 00:16:45.030
And I apologize because I
realize when I was trying to

00:16:45.030 --> 00:16:48.060
explain it without the example,
it might have gotten

00:16:48.060 --> 00:16:48.740
a little confusing.

00:16:48.740 --> 00:16:52.130
But hopefully this example
explains things

00:16:52.130 --> 00:16:52.890
a little bit better.

00:16:52.890 --> 00:16:55.410
Although it might be confusing
to see green and purple

00:16:55.410 --> 00:16:56.480
written in blue.

00:16:56.480 --> 00:16:58.200
See you in the next video.