Discrete and continuous random variables

WEBVTT
Kind: captions
Language: en

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We already know a little
bit about random variables.

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What we're going to
see in this video

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is that random variables
come in two varieties.

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You have discrete
random variables,

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and you have continuous
random variables.

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And discrete random
variables, these

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are essentially
random variables that

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can take on distinct
or separate values.

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And we'll give examples
of that in a second.

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So that comes straight from the
meaning of the word discrete

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in the English language--
distinct or separate values.

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While continuous-- and I
guess just another definition

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for the word discrete
in the English language

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would be polite, or not
obnoxious, or kind of subtle.

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That is not what
we're talking about.

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We are not talking about random
variables that are polite.

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We're talking about ones that
can take on distinct values.

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And continuous random
variables, they

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can take on any
value in a range.

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And that range could
even be infinite.

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So any value in an interval.

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So with those two
definitions out of the way,

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let's look at some actual
random variable definitions.

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And I want to think together
about whether you would

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classify them as discrete or
continuous random variables.

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So let's say that I have a
random variable capital X.

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And it is equal to--
well, this is one

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that we covered
in the last video.

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It's 1 if my fair coin is heads.

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It's 0 if my fair coin is tails.

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So is this a discrete or a
continuous random variable?

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Well, this random
variable right over here

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can take on distinctive values.

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It can take on either a 1
or it could take on a 0.

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Another way to think
about it is you

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can count the number
of different values

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it can take on.

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This is the first
value it can take on,

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this is the second value
that it can take on.

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So this is clearly a
discrete random variable.

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Let's think about another one.

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Let's define random
variable Y as

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equal to the mass of a random
animal selected at the New

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Orleans zoo, where I
grew up, the Audubon Zoo.

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Y is the mass of a random animal
selected at the New Orleans

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zoo.

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Is this a discrete
random variable

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or a continuous random variable?

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Well, the exact mass--
and I should probably

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put that qualifier here.

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I'll even add it here just to
make it really, really clear.

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The exact mass of a random
animal, or a random object

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in our universe, it can take on
any of a whole set of values.

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I mean, who knows
exactly the exact number

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of electrons that are
part of that object

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right at that moment?

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Who knows the
neutrons, the protons,

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the exact number of
molecules in that object,

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or a part of that animal
exactly at that moment?

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So that mass, for
example, at the zoo,

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it might take on a value
anywhere between-- well,

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maybe close to 0.

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There's no animal
that has 0 mass.

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But it could be close to zero,
if we're thinking about an ant,

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or we're thinking
about a dust mite,

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or maybe if you consider
even a bacterium an animal.

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I believe bacterium is
the singular of bacteria.

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And it could go all the way.

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Maybe the most massive
animal in the zoo

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is the elephant of some kind.

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And I don't know what it
would be in kilograms,

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but it would be fairly large.

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So maybe you can
get up all the way

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to 3,000 kilograms,
or probably larger.

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Let's say 5,000 kilograms.

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I don't know what the mass of a
very heavy elephant-- or a very

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massive elephant, I
should say-- actually is.

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It may be something
fun for you to look at.

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But any animal could have a
mass anywhere in between here.

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It does not take
on discrete values.

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You could have an animal that
is exactly maybe 123.75921

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kilograms.

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And even there, that actually
might not be the exact mass.

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You might have to get even
more precise, --10732.

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0, 7, And I think
you get the picture.

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Even though this is the
way I've defined it now,

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a finite interval, you can take
on any value in between here.

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They are not discrete values.

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So this one is clearly a
continuous random variable.

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Let's think about another one.

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Let's think about-- let's say
that random variable Y, instead

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of it being this, let's say it's
the year that a random student

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in the class was born.

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Is this a discrete or a
continuous random variable?

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Well, that year, you
literally can define it

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as a specific discrete year.

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It could be 1992, or it could
be 1985, or it could be 2001.

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There are discrete values
that this random variable

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can actually take on.

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It won't be able to take on
any value between, say, 2000

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and 2001.

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It'll either be 2000 or
it'll be 2001 or 2002.

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Once again, you can count
the values it can take on.

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Most of the times that
you're dealing with,

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as in the case right here,
a discrete random variable--

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let me make it clear
this one over here

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is also a discrete
random variable.

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Most of the time
that you're dealing

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with a discrete random
variable, you're

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probably going to be dealing
with a finite number of values.

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But it does not have to be
a finite number of values.

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You can actually have an
infinite potential number

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of values that it
could take on--

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as long as the
values are countable.

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As long as you
can literally say,

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OK, this is the first
value it could take on,

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the second, the third.

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And you might be counting
forever, but as long

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as you can literally
list-- and it

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could be even an infinite list.

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But if you can list the
values that it could take on,

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then you're dealing with a
discrete random variable.

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Notice in this
scenario with the zoo,

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you could not list all
of the possible masses.

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You could not even count them.

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You might attempt to--
and it's a fun exercise

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to try at least
once, to try to list

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all of the values
this might take on.

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You might say,
OK, maybe it could

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take on 0.01 and maybe 0.02.

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But wait, you just skipped
an infinite number of values

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that it could take on, because
it could have taken on 0.011,

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0.012.

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And even between those,
there's an infinite number

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of values it could take on.

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There's no way for you to
count the number of values

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that a continuous random
variable can take on.

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There's no way for
you to list them.

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With a discrete random variable,
you can count the values.

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You can list the values.

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Let's do another example.

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Let's let random
variable Z, capital Z,

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be the number ants born
tomorrow in the universe.

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Now, you're probably
arguing that there

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aren't ants on other planets.

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Or maybe there are
ant-like creatures,

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but they're not going to
be ants as we define them.

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But how do we know?

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So number of ants
born in the universe.

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Maybe some ants have figured
out interstellar travel

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of some kind.

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So the number of ants born
tomorrow in the universe.

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That's my random variable Z. Is
this a discrete random variable

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or a continuous random variable?

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Well, once again, we
can count the number

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of values this could take on.

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This could be 1.

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It could be 2.

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It could be 3.

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It could be 4.

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It could be 5 quadrillion ants.

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It could be 5 quadrillion and 1.

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We can actually
count the values.

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Those values are discrete.

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So once again, this
right over here

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is a discrete random variable.

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This is fun, so let's
keep doing more of these.

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Let's say that I have
random variable X.

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So we're not using this
definition anymore.

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Now I'm going to define
random variable X

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to be the winning time-- now
let me write it this way.

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The exact winning time for
the men's 100-meter dash

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at the 2016 Olympics.

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So the exact time that it took
for the winner-- who's probably

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going to be Usain Bolt,
but it might not be.

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Actually, he's
aging a little bit.

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But whatever the exact
winning time for the men's

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100-meter in the 2016 Olympics.

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And not the one that you
necessarily see on the clock.

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The exact, the
precise time that you

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would see at the
men's 100-meter dash.

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Is this a discrete or a
continuous random variable?

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Well, the way I've defined, and
this one's a little bit tricky.

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Because you might
say it's countable.

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You might say, well,
it could either

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be 956, 9.56 seconds, or 9.57
seconds, or 9.58 seconds.

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And you might be
tempted to believe

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that, because when you watch the
100-meter dash at the Olympics,

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they measure it to the
nearest hundredths.

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They round to the
nearest hundredth.

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That's how precise
their timing is.

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But I'm talking about the exact
winning time, the exact number

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of seconds it takes
for that person

00:10:02.610 --> 00:10:06.000
to, from the starting gun,
to cross the finish line.

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And there, it can
take on any value.

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It can take on any
value between-- well,

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I guess they're limited
by the speed of light.

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But it could take on any
value you could imagine.

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It might be anywhere between 5
seconds and maybe 12 seconds.

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And it could be anywhere
in between there.

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It might not be 9.57.

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That might be what
the clock says,

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but in reality the exact
winning time could be 9.571,

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or it could be 9.572359.

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I think you see what I'm saying.

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The exact precise time could
be any value in an interval.

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So this right over here is a
continuous random variable.

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Now what would be
the case, instead

00:10:52.090 --> 00:10:53.880
of saying the
exact winning time,

00:10:53.880 --> 00:10:58.970
if instead I defined X to be the
winning time of the men's 100

00:10:58.970 --> 00:11:09.051
meter dash at the 2016
Olympics rounded to the nearest

00:11:09.051 --> 00:11:09.550
hundredth?

00:11:12.480 --> 00:11:15.494
Is this a discrete or a
continuous random variable?

00:11:15.494 --> 00:11:16.410
So let me delete this.

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I've changed the
random variable now.

00:11:18.490 --> 00:11:20.090
Is this going to
be a discrete or

00:11:20.090 --> 00:11:22.810
a continuous random variable?

00:11:22.810 --> 00:11:25.987
Well now, we can actually
count the actual values

00:11:25.987 --> 00:11:27.570
that this random
variable can take on.

00:11:27.570 --> 00:11:29.270
It might be 9.56.

00:11:29.270 --> 00:11:30.780
It could be 9.57.

00:11:30.780 --> 00:11:32.370
It could be 9.58.

00:11:32.370 --> 00:11:34.540
We can actually list them.

00:11:34.540 --> 00:11:37.109
So in this case, when we round
it to the nearest hundredth,

00:11:37.109 --> 00:11:38.400
we can actually list of values.

00:11:38.400 --> 00:11:47.670
We are now dealing with a
discrete random variable.

00:11:47.670 --> 00:11:49.230
Anyway, I'll let you go there.

00:11:49.230 --> 00:11:52.090
Hopefully this gives you
a sense of the distinction

00:11:52.090 --> 00:11:56.330
between discrete and
continuous random variables.


Random variables

WEBVTT
Kind: captions
Language: en

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What I want to discuss a
little bit in this video

00:00:03.310 --> 00:00:08.970
is the idea of a
random variable.

00:00:08.970 --> 00:00:11.930
And random variables at first
can be a little bit confusing

00:00:11.930 --> 00:00:15.000
because we will want to think
of them as traditional variables

00:00:15.000 --> 00:00:17.380
that you were first exposed
to in algebra class.

00:00:17.380 --> 00:00:20.160
And that's not quite what
random variables are.

00:00:20.160 --> 00:00:23.280
Random variables are
really ways to map outcomes

00:00:23.280 --> 00:00:25.490
of random processes to numbers.

00:00:25.490 --> 00:00:29.320
So if you have a random process,
like you're flipping a coin

00:00:29.320 --> 00:00:34.090
or you're rolling dice or you
are measuring the rain that

00:00:34.090 --> 00:00:37.240
might fall tomorrow,
so random process,

00:00:37.240 --> 00:00:44.735
you're really just mapping
outcomes of that to numbers.

00:00:44.735 --> 00:00:49.340
You are quantifying
the outcomes.

00:00:49.340 --> 00:00:51.640
So what's an example
of a random variable?

00:00:51.640 --> 00:00:53.270
Well, let's define
one right over here.

00:00:53.270 --> 00:00:55.940
So I'm going to define
random variable capital X.

00:00:55.940 --> 00:00:59.290
And they tend to be
denoted by capital letters.

00:00:59.290 --> 00:01:01.730
So random variable capital
X, I will define it

00:01:01.730 --> 00:01:05.410
as-- It is going
to be equal to 1

00:01:05.410 --> 00:01:11.100
if my fair die rolls heads--
let me write it this way--

00:01:11.100 --> 00:01:11.615
if heads.

00:01:16.090 --> 00:01:20.464
And it's going to be
equal to 0 if tails.

00:01:20.464 --> 00:01:22.380
I could have defined
this any way I wanted to.

00:01:22.380 --> 00:01:23.921
This is actually a
fairly typical way

00:01:23.921 --> 00:01:26.690
of defining a random variable,
especially for a coin flip.

00:01:26.690 --> 00:01:28.670
But I could have
defined this as 100.

00:01:28.670 --> 00:01:31.450
And I could have
defined this as 703.

00:01:31.450 --> 00:01:34.710
And this would still be a
legitimate random variable.

00:01:34.710 --> 00:01:38.030
It might not be as pure a
way of thinking about it

00:01:38.030 --> 00:01:40.500
as defining 1 as
heads and 0 as tails.

00:01:40.500 --> 00:01:42.840
But that would have
been a random variable.

00:01:42.840 --> 00:01:47.000
Notice we have taken this
random process, flipping a coin,

00:01:47.000 --> 00:01:49.727
and we've mapped the outcomes
of that random process.

00:01:49.727 --> 00:01:50.810
And we've quantified them.

00:01:50.810 --> 00:01:53.750
1 if heads, 0 if tails.

00:01:53.750 --> 00:02:01.000
We can define another random
variable capital Y as equal to,

00:02:01.000 --> 00:02:10.569
let's say, the sum of
rolls of let's say 7 dice.

00:02:10.569 --> 00:02:12.110
And when we talk
about the sum, we're

00:02:12.110 --> 00:02:14.470
talking about the
sum of the 7-- let

00:02:14.470 --> 00:02:26.520
me write this-- the
sum of the upward face

00:02:26.520 --> 00:02:32.960
after rolling 7 dice.

00:02:32.960 --> 00:02:37.730
Once again, we are quantifying
an outcome for a random process

00:02:37.730 --> 00:02:41.030
where the random process
is rolling these 7 dice

00:02:41.030 --> 00:02:43.820
and seeing what
sides show up on top.

00:02:43.820 --> 00:02:46.260
And then we are taking those
and we're taking the sum

00:02:46.260 --> 00:02:51.710
and we are defining a
random variable in that way.

00:02:51.710 --> 00:02:53.490
So the natural
question you might ask

00:02:53.490 --> 00:02:55.100
is, why are we doing this?

00:02:55.100 --> 00:02:58.680
What's so useful about defining
random variables like this?

00:02:58.680 --> 00:03:00.310
It will become
more apparent as we

00:03:00.310 --> 00:03:02.206
get a little bit
deeper in probability.

00:03:02.206 --> 00:03:03.830
But the simple way
of thinking about it

00:03:03.830 --> 00:03:06.710
is as soon as you
quantify outcomes,

00:03:06.710 --> 00:03:10.529
you can start to do a little
bit more math on the outcomes.

00:03:10.529 --> 00:03:12.070
And you can start
to use a little bit

00:03:12.070 --> 00:03:15.850
more mathematical
notation on the outcome.

00:03:15.850 --> 00:03:18.380
So for example, if you
cared about the probability

00:03:18.380 --> 00:03:21.430
that the sum of the upward
faces after rolling seven

00:03:21.430 --> 00:03:23.200
dice-- if you cared
about the probability

00:03:23.200 --> 00:03:27.532
that that sum is less than
or equal to 30, the old way

00:03:27.532 --> 00:03:28.990
that you would have
to have written

00:03:28.990 --> 00:03:31.350
it is the probability
that the sum

00:03:31.350 --> 00:03:39.150
of-- and you would have to write
all of what I just wrote here--

00:03:39.150 --> 00:03:43.600
is less than or equal to 30.

00:03:43.600 --> 00:03:46.090
You would have had to
write that big thing.

00:03:46.090 --> 00:03:48.380
And then you would try
to figure it out somehow

00:03:48.380 --> 00:03:49.910
if you had some information.

00:03:49.910 --> 00:03:53.010
But now we can just
write the probability

00:03:53.010 --> 00:03:57.330
that capital Y is less
than or equal to 30.

00:03:57.330 --> 00:03:59.370
It's a little bit
cleaner notation.

00:03:59.370 --> 00:04:03.130
And if someone else cares
about the probability

00:04:03.130 --> 00:04:06.600
that this sum of the upward
face after rolling seven dice--

00:04:06.600 --> 00:04:09.112
if they say, hey, what's the
probability that that's even,

00:04:09.112 --> 00:04:10.820
instead of having to
write all that over,

00:04:10.820 --> 00:04:17.839
they can say, well, what's the
probability that Y is even?

00:04:17.839 --> 00:04:19.890
Now the one thing that
I do want to emphasize

00:04:19.890 --> 00:04:22.460
is how these are different
than traditional variables,

00:04:22.460 --> 00:04:24.860
traditional variables that
you see in your algebra class

00:04:24.860 --> 00:04:27.420
like x plus 5 is equal
to 6, usually denoted

00:04:27.420 --> 00:04:29.180
by lowercase variables.

00:04:29.180 --> 00:04:32.330
y is equal to x plus 7.

00:04:32.330 --> 00:04:35.440
These variables, you can
essentially assign values.

00:04:35.440 --> 00:04:38.350
You either can solve for
them-- so in this case,

00:04:38.350 --> 00:04:39.360
x is an unknown.

00:04:39.360 --> 00:04:42.120
You could subtract 5 from
both sides and solve for x.

00:04:42.120 --> 00:04:44.990
Say that x is going
to be equal to 1.

00:04:44.990 --> 00:04:48.290
In this case, you could say,
well, x is going to vary.

00:04:48.290 --> 00:04:50.690
We can assign a
value to x and see

00:04:50.690 --> 00:04:53.390
how y varies as a function of x.

00:04:53.390 --> 00:04:54.770
You can either
assign a variable,

00:04:54.770 --> 00:04:56.510
you can assign values to them.

00:04:56.510 --> 00:04:58.950
Or you can solve for them.

00:04:58.950 --> 00:05:02.780
You could say, hey x is
going to be 1 in this case.

00:05:02.780 --> 00:05:05.360
That's not going to be the
case with a random variable.

00:05:05.360 --> 00:05:07.930
A random variable can take on
many, many, many, many, many,

00:05:07.930 --> 00:05:10.660
many different values with
different probabilities.

00:05:10.660 --> 00:05:12.320
And it makes much
more sense to talk

00:05:12.320 --> 00:05:15.480
about the probability of
a random variable equaling

00:05:15.480 --> 00:05:18.430
a value, or the probability
that it is less than or greater

00:05:18.430 --> 00:05:20.140
than something,
or the probability

00:05:20.140 --> 00:05:21.990
that it has some property.

00:05:21.990 --> 00:05:24.669
And you see that in
either of these cases.

00:05:24.669 --> 00:05:26.710
In the next video, we'll
continue this discussion

00:05:26.710 --> 00:05:28.670
and we'll talk a little
bit about the types

00:05:28.670 --> 00:05:31.650
of random variables
you can have.


Dependent probability introduction

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.420
Let's imagine
ourselves in some type

00:00:02.420 --> 00:00:05.510
of a strange casino
with very strange games.

00:00:05.510 --> 00:00:07.900
And you walk up to a
table, and on that table

00:00:07.900 --> 00:00:10.660
there is an empty bag.

00:00:10.660 --> 00:00:12.990
And the guy who
runs the table says,

00:00:12.990 --> 00:00:15.630
look, I've got
some marbles here,

00:00:15.630 --> 00:00:18.055
3 green marbles,
2 orange marbles.

00:00:18.055 --> 00:00:19.680
And I'm going to
stick them in the bag.

00:00:19.680 --> 00:00:22.120
And he literally sticks
them into the empty bag

00:00:22.120 --> 00:00:26.840
to show you that it's
truly 3 green marbles and 2

00:00:26.840 --> 00:00:27.535
orange marbles.

00:00:30.930 --> 00:00:33.700
And he says, the game
that I want you to play,

00:00:33.700 --> 00:00:36.470
or if you choose
to play, is you're

00:00:36.470 --> 00:00:38.590
going to look away-- stick
your hand in this bag,

00:00:38.590 --> 00:00:41.000
the bag is not transparent--
feel around the marbles.

00:00:41.000 --> 00:00:43.170
All the marbles feel
exactly the same.

00:00:43.170 --> 00:00:46.070
And if you're able to
pick 2 green marbles,

00:00:46.070 --> 00:00:49.230
if you're able to take 1 marble
out of the bag, it's green.

00:00:49.230 --> 00:00:50.500
You put it down on the table.

00:00:50.500 --> 00:00:53.830
Then put your hand back in the
bag, and take another marble.

00:00:53.830 --> 00:00:56.180
And if that one's also
green, then you're

00:00:56.180 --> 00:00:59.350
going to win the prize.

00:00:59.350 --> 00:01:04.500
You're going to win $1
if you get 2 greens.

00:01:04.500 --> 00:01:06.870
If you get 2 greens,
you're going to win $1.

00:01:06.870 --> 00:01:09.090
You say, well, this sounds
like an interesting game.

00:01:09.090 --> 00:01:10.910
How much does it cost to play?

00:01:10.910 --> 00:01:15.270
And the guy tells you,
it is $0.35 to play,

00:01:15.270 --> 00:01:19.240
so obviously a fairly
low stakes casino.

00:01:19.240 --> 00:01:23.490
So my question to you is, would
you want to play this game?

00:01:23.490 --> 00:01:25.410
And don't put the
fun factor into it.

00:01:25.410 --> 00:01:28.630
Just economically, does
it make sense for you

00:01:28.630 --> 00:01:32.810
to actually play this game?

00:01:32.810 --> 00:01:36.450
Well, let's think through the
probabilities a little bit.

00:01:36.450 --> 00:01:39.090
So, first of all,
what's the probability

00:01:39.090 --> 00:01:41.155
that the first marble
you pick is green?

00:01:48.052 --> 00:01:49.760
Actually, let me just
write, first green,

00:01:49.760 --> 00:01:51.240
probability first green.

00:01:54.730 --> 00:01:57.040
Well, the total possible
outcomes-- there's

00:01:57.040 --> 00:01:59.590
5 marbles here,
all equally likely.

00:01:59.590 --> 00:02:01.590
So there's 5 possible outcomes.

00:02:01.590 --> 00:02:04.580
3 of them satisfy your event
that the first is green.

00:02:04.580 --> 00:02:08.639
So there's a 3/5 probability
that the first is green.

00:02:08.639 --> 00:02:12.380
So you have a 3/5 chance,
3/5 probability I should say,

00:02:12.380 --> 00:02:14.640
that after that first
pick you're kind of still

00:02:14.640 --> 00:02:16.060
in the game.

00:02:16.060 --> 00:02:19.870
Now, what we really care
about is your probability

00:02:19.870 --> 00:02:21.620
of winning the game.

00:02:21.620 --> 00:02:25.270
You want the first to be
green, and the second green.

00:02:25.270 --> 00:02:27.020
Well, let's think about
this a little bit.

00:02:27.020 --> 00:02:31.590
What is the probability that
the first is green-- first, I'll

00:02:31.590 --> 00:02:39.310
just write g for green--
and the second is green?

00:02:39.310 --> 00:02:41.630
Now, you might be
tempted to say, oh, well,

00:02:41.630 --> 00:02:44.080
maybe the second being green
is the same probability.

00:02:44.080 --> 00:02:45.200
It's 3/5.

00:02:45.200 --> 00:02:47.810
I can just multiply
3/5 times 3/5,

00:02:47.810 --> 00:02:52.350
and I'll get 9/25, seems like
a pretty straightforward thing.

00:02:52.350 --> 00:02:54.330
But the realization
here is what you

00:02:54.330 --> 00:02:56.420
do with that first green marble.

00:02:56.420 --> 00:02:59.020
You don't take that first
green marble out, look at it,

00:02:59.020 --> 00:03:00.920
and put it back in the bag.

00:03:00.920 --> 00:03:03.790
So when you take
that second pick,

00:03:03.790 --> 00:03:05.801
the number of green
marbles that are in the bag

00:03:05.801 --> 00:03:07.550
depends on what you
got on the first pick.

00:03:07.550 --> 00:03:09.550
Remember, we take
the marble out.

00:03:09.550 --> 00:03:11.360
If it's a green
marble, whatever marble

00:03:11.360 --> 00:03:13.050
it is, at whatever
after the first pick,

00:03:13.050 --> 00:03:14.480
we leave it on the table.

00:03:14.480 --> 00:03:15.990
We are not replacing it.

00:03:15.990 --> 00:03:17.960
So there's not any
replacement here.

00:03:17.960 --> 00:03:20.560
So these are not
independent events.

00:03:20.560 --> 00:03:24.100
Let me make this
clear, not independent.

00:03:24.100 --> 00:03:31.510
Or in particular, the second
pick is dependent on the first.

00:03:31.510 --> 00:03:36.920
Dependent on the first pick.

00:03:36.920 --> 00:03:39.370
If the first pick
is green, then you

00:03:39.370 --> 00:03:42.130
don't have 3 green
marbles in a bag of 5.

00:03:42.130 --> 00:03:44.120
If the first pick
is green, you now

00:03:44.120 --> 00:03:48.200
have 2 green marbles
in a bag of 4.

00:03:48.200 --> 00:03:49.829
So the way that we
would refer to this,

00:03:49.829 --> 00:03:51.620
the probability of both
of these happening,

00:03:51.620 --> 00:03:53.830
yes, it's definitely
equal to the probability

00:03:53.830 --> 00:04:02.500
of the first green times-- now,
this is kind of the new idea--

00:04:02.500 --> 00:04:09.260
the probability of the second
green given-- this little line

00:04:09.260 --> 00:04:11.750
right over here, just a
straight up vertical line, just

00:04:11.750 --> 00:04:19.000
means given-- given that
the first was green.

00:04:19.000 --> 00:04:22.570
Now, what is the probability
that the second marble is green

00:04:22.570 --> 00:04:26.270
given that the first
marble was green?

00:04:26.270 --> 00:04:28.490
Well, we drew the
scenario right over here.

00:04:28.490 --> 00:04:31.280
If the first marble
is green, there

00:04:31.280 --> 00:04:36.570
are 4 possible outcomes,
not 5 anymore, and 2 of them

00:04:36.570 --> 00:04:39.150
satisfy your criteria.

00:04:39.150 --> 00:04:41.280
So 2 of them satisfy
your criteria.

00:04:41.280 --> 00:04:43.830
So the probability of the
first marble green being green

00:04:43.830 --> 00:04:45.997
and the second
marble being green,

00:04:45.997 --> 00:04:48.330
is going to be the probability
that your first is green,

00:04:48.330 --> 00:04:51.840
so it's going to be 3/5,
times the probability

00:04:51.840 --> 00:04:54.390
that the second is green,
given that the first was green.

00:04:54.390 --> 00:04:56.700
Now you have 1 less
marble in the bag,

00:04:56.700 --> 00:04:58.930
and we're assuming that
the first pick was green,

00:04:58.930 --> 00:05:01.970
so you'll only have
2 green marbles left.

00:05:01.970 --> 00:05:05.280
And so what does this give
us for our total probability?

00:05:05.280 --> 00:05:07.025
Let's see, 3/5 times 2/4.

00:05:07.025 --> 00:05:09.480
Well, 2/4 is the
same thing as 1/2.

00:05:09.480 --> 00:05:13.100
This is going to be
equal to 3/5 times

00:05:13.100 --> 00:05:16.690
1/2, which is equal to 3/10.

00:05:16.690 --> 00:05:24.720
Or we could write that as 0.30,
or we could say there's a 30%

00:05:24.720 --> 00:05:27.920
chance of picking
2 green marbles,

00:05:27.920 --> 00:05:30.000
when we are not replacing.

00:05:30.000 --> 00:05:32.440
So given that, let me ask
you the question again.

00:05:32.440 --> 00:05:35.782
Would you want to
play this game?

00:05:35.782 --> 00:05:37.990
Well, if you played this
game many, many, many, many,

00:05:37.990 --> 00:05:47.500
many times, on average you have
a 30% chance of winning $1.

00:05:47.500 --> 00:05:51.080
And we haven't covered this yet,
but so your expected value is

00:05:51.080 --> 00:05:55.500
really going to
be 30% times $1--

00:05:55.500 --> 00:05:57.320
this gives you a little
bit of a preview--

00:05:57.320 --> 00:06:03.230
which is going to be $0.30
30% chance of winning $1.

00:06:03.230 --> 00:06:05.790
You would expect on average,
if you were to play this many,

00:06:05.790 --> 00:06:09.410
many, many times, that playing
the game is going to give you

00:06:09.410 --> 00:06:10.940
$0.30.

00:06:10.940 --> 00:06:16.090
Now, would you want to
give someone $0.35 to get,

00:06:16.090 --> 00:06:17.910
on average, $0.30?

00:06:17.910 --> 00:06:21.620
No, you would not want
to play this game.

00:06:21.620 --> 00:06:23.470
Now, one thing I will
let you think about

00:06:23.470 --> 00:06:27.510
is, would you want to play
this game if you could replace

00:06:27.510 --> 00:06:29.420
the green marble,
the first pick.

00:06:29.420 --> 00:06:33.650
After the first pick, if you
could replace the green marble,

00:06:33.650 --> 00:06:36.863
would you want to play
the game in that scenario?


More fiscal cliff analysis

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.310
In the last video, we
gave a basic outline

00:00:03.310 --> 00:00:06.310
of the different scenarios
that might play out in 2013.

00:00:06.310 --> 00:00:08.670
Or at least the different
budgetary proposals

00:00:08.670 --> 00:00:10.750
on the table from
the administration,

00:00:10.750 --> 00:00:12.570
from the Republicans,
and what will

00:00:12.570 --> 00:00:14.450
happen if they don't
come into agreement,

00:00:14.450 --> 00:00:15.730
which was the fiscal cliff.

00:00:15.730 --> 00:00:18.510
Which is this fairly
unusual situation

00:00:18.510 --> 00:00:20.700
where the both parties
to the negotiation

00:00:20.700 --> 00:00:22.850
set up this thing that
will automatically

00:00:22.850 --> 00:00:25.410
happen at the beginning of 2013.

00:00:25.410 --> 00:00:27.730
That it would be painful
for both parties,

00:00:27.730 --> 00:00:29.770
and the thinking
being that it'll

00:00:29.770 --> 00:00:32.590
force them to come, maybe,
to some type of an agreement

00:00:32.590 --> 00:00:33.450
sooner than later.

00:00:33.450 --> 00:00:35.610
The fiscal cliff is what
neither the Republicans

00:00:35.610 --> 00:00:39.070
or the administration want
because it raises taxes, which

00:00:39.070 --> 00:00:42.680
the Republicans don't want,
and, it reduces spending,

00:00:42.680 --> 00:00:44.350
which the president
doesn't want.

00:00:44.350 --> 00:00:46.976
And the core argument, and we
touched it on the last video

00:00:46.976 --> 00:00:48.600
and we'll go in more
depth in this one,

00:00:48.600 --> 00:00:51.330
is just as we're
beginning to recover

00:00:51.330 --> 00:00:53.790
from our last
recession, it might not

00:00:53.790 --> 00:00:58.360
be useful for our economy to
try to suck out $1/2 trillion.

00:00:58.360 --> 00:01:02.970
So let's think about the
different issues at play.

00:01:02.970 --> 00:01:05.599
So right over here,
we have some charts.

00:01:05.599 --> 00:01:07.890
Most of these are from the
Congressional Budget Office.

00:01:07.890 --> 00:01:10.220
And there's a bunch of
assumptions that go into this.

00:01:10.220 --> 00:01:12.270
And there many people
will debate the assumption

00:01:12.270 --> 00:01:13.000
that they make.

00:01:13.000 --> 00:01:17.900
But they at least directionally
show the right things.

00:01:17.900 --> 00:01:20.650
So this right over
here, this first chart,

00:01:20.650 --> 00:01:22.940
compares deficits or surpluses.

00:01:22.940 --> 00:01:25.370
And really, ever
since the early 2000s,

00:01:25.370 --> 00:01:27.610
we've been running deficits.

00:01:27.610 --> 00:01:31.140
And as we got into
the late, I guess,

00:01:31.140 --> 00:01:35.610
'10s, or 2008, 2010
timeframe, 2009 timeframe.

00:01:35.610 --> 00:01:37.760
As we went into a
financial crisis.

00:01:37.760 --> 00:01:41.280
When the economy
contracts, you get

00:01:41.280 --> 00:01:44.110
hit in two ways when you
think about deficit spending.

00:01:44.110 --> 00:01:46.290
On one end, you bring
in less revenues.

00:01:46.290 --> 00:01:47.660
The economy is shrinking.

00:01:47.660 --> 00:01:49.150
And on the other end,
you have to spend more.

00:01:49.150 --> 00:01:50.733
You have to give
people more benefits.

00:01:50.733 --> 00:01:51.900
More unemployment benefits.

00:01:51.900 --> 00:01:52.900
Things like that.

00:01:52.900 --> 00:01:55.670
So whenever you see the
economy shrink, you will see,

00:01:55.670 --> 00:01:59.920
naturally, all other
things being equal,

00:01:59.920 --> 00:02:01.460
you will see deficits increase.

00:02:01.460 --> 00:02:02.830
And that's what you
saw right over here.

00:02:02.830 --> 00:02:04.246
And on top of that,
the government

00:02:04.246 --> 00:02:07.180
is trying to bail things out as
doing stimulus spending and all

00:02:07.180 --> 00:02:09.550
that in order to minimize
the effect of the deficit.

00:02:09.550 --> 00:02:12.840
And that's why you see
here in 2008, 2009,

00:02:12.840 --> 00:02:16.020
we start running
significant deficits.

00:02:16.020 --> 00:02:18.940
Now, what's
interesting is what's

00:02:18.940 --> 00:02:21.710
going to happen going forward.

00:02:21.710 --> 00:02:24.740
And going forward, there's two
of these scenarios right here.

00:02:24.740 --> 00:02:28.900
There is the CBO's
baseline projection

00:02:28.900 --> 00:02:31.930
and then there's the
alternative fiscal scenario.

00:02:31.930 --> 00:02:34.210
And just to be
clear where we, we

00:02:34.210 --> 00:02:40.550
are entering this phase
right over here in 2013.

00:02:40.550 --> 00:02:43.640
And the baseline
projection is essentially

00:02:43.640 --> 00:02:46.020
if the government
takes no more action.

00:02:46.020 --> 00:02:48.260
And if the government
takes no more action,

00:02:48.260 --> 00:02:50.870
then the fiscal cliff
will be triggered.

00:02:50.870 --> 00:02:56.640
We will essentially have $500
billion less in 2013's deficit.

00:02:56.640 --> 00:02:58.750
And so what you see here
is that the deficits

00:02:58.750 --> 00:02:59.760
go down dramatically.

00:03:03.620 --> 00:03:07.230
We have to borrow less
on an annual basis.

00:03:07.230 --> 00:03:09.990
Now, the alternative
to fiscal scenario

00:03:09.990 --> 00:03:14.930
is if we essentially continue--
if we extend our existing

00:03:14.930 --> 00:03:19.940
spending, and if we continue to
extend the Bush era tax cuts,

00:03:19.940 --> 00:03:22.250
we go along this path.

00:03:22.250 --> 00:03:24.940
And so if you care a lot about
deficits, the fiscal cliff

00:03:24.940 --> 00:03:26.684
scenario, which
is this top line.

00:03:26.684 --> 00:03:29.100
This is actually the fiscal--
let me write that in a color

00:03:29.100 --> 00:03:33.090
you can see-- that's actually
the fiscal cliff scenario.

00:03:33.090 --> 00:03:34.340
It actually looks pretty good.

00:03:34.340 --> 00:03:38.410
Hey, look, we have much, much
lower deficits in that scenario

00:03:38.410 --> 00:03:43.810
than if we were to
continue the tax cuts,

00:03:43.810 --> 00:03:46.000
and if we were to
continue the spending.

00:03:46.000 --> 00:03:47.960
So there that looks
like, actually,

00:03:47.960 --> 00:03:50.300
an argument for
the fiscal cliff.

00:03:50.300 --> 00:03:51.491
Now let's look at this.

00:03:51.491 --> 00:03:53.990
Now, this is-- we're going to
talk about the aggregate debt.

00:03:53.990 --> 00:03:55.500
When we say deficit,
we're talking

00:03:55.500 --> 00:03:58.470
about the shortfall
in a given year.

00:03:58.470 --> 00:04:01.750
The debt is the aggregate
amount that the government owes.

00:04:01.750 --> 00:04:04.810
So this is federal debt
held by the public,

00:04:04.810 --> 00:04:08.310
historically and as
projected in CBO's baseline.

00:04:08.310 --> 00:04:11.060
And CBO, when they
project, they can only

00:04:11.060 --> 00:04:14.270
do so much in projecting how
good the economy might be,

00:04:14.270 --> 00:04:15.250
or how bad it might be.

00:04:15.250 --> 00:04:16.950
To a large degree,
they extrapolate

00:04:16.950 --> 00:04:19.310
from where we are right now.

00:04:19.310 --> 00:04:23.130
And right over here, you see
debt as a percentage of GDP.

00:04:23.130 --> 00:04:25.430
So you could imagine
during World War II,

00:04:25.430 --> 00:04:27.560
we had a lot of debt
as a percentage of GDP.

00:04:27.560 --> 00:04:30.810
See over here after the
end of World War II,

00:04:30.810 --> 00:04:34.660
we start crossing over
100% of GDP as our debt--

00:04:34.660 --> 00:04:37.620
the aggregate amount
that the government owes.

00:04:37.620 --> 00:04:38.980
And then it delevered.

00:04:38.980 --> 00:04:42.080
And then you see this
rough trend, roughly

00:04:42.080 --> 00:04:46.150
from the early '80s
until now, where

00:04:46.150 --> 00:04:50.240
we've had increasing
aggregate debt.

00:04:50.240 --> 00:04:55.240
There were some moments in the
late '90s, early 2000s, where

00:04:55.240 --> 00:04:58.250
we took it down-- very,
very, very strong economy.

00:04:58.250 --> 00:04:59.500
We started running surpluses.

00:04:59.500 --> 00:05:01.580
We started to pay
down some of our debt.

00:05:01.580 --> 00:05:05.670
But then we hit back on this
debt increasing trend line.

00:05:05.670 --> 00:05:08.870
And once again,
if we continue in

00:05:08.870 --> 00:05:11.400
the alternative fiscal scenario,
which is really continuing

00:05:11.400 --> 00:05:14.370
to do what we do today,
which is extend the tax cuts

00:05:14.370 --> 00:05:17.640
and continue to have roughly
the same level of spending,

00:05:17.640 --> 00:05:20.810
you see that the debt only
continues to increase.

00:05:23.380 --> 00:05:25.080
In the baseline
projection-- this

00:05:25.080 --> 00:05:26.760
is actually the
fiscal cliff scenario.

00:05:26.760 --> 00:05:29.490
This is where no new
legislation has passed.

00:05:29.490 --> 00:05:33.350
Those triggers in which the
Bush era tax cuts expire,

00:05:33.350 --> 00:05:38.020
and the spending cuts that are
triggered by the fiscal cliff

00:05:38.020 --> 00:05:40.500
scenario hit, we
run lower deficits.

00:05:40.500 --> 00:05:44.520
And we actually see that
debt, as a percentage of GDP,

00:05:44.520 --> 00:05:46.590
actually starts to go down.

00:05:46.590 --> 00:05:50.160
Which is, at least from a
balance sheet prospective,

00:05:50.160 --> 00:05:51.164
seems like a good thing.

00:05:51.164 --> 00:05:53.580
So these first two charts say,
hey maybe this fiscal cliff

00:05:53.580 --> 00:05:54.830
thing isn't so bad.

00:05:54.830 --> 00:05:57.760
It will actually
significantly lower

00:05:57.760 --> 00:06:00.060
the deficit on an annual basis.

00:06:00.060 --> 00:06:05.352
And as a percentage of GDP,
we will as a country, delever.

00:06:05.352 --> 00:06:06.810
Now let's think
about the arguments

00:06:06.810 --> 00:06:09.790
why well, maybe we don't want
to be that aggressive when

00:06:09.790 --> 00:06:12.650
it comes to reducing
our deficit,

00:06:12.650 --> 00:06:16.280
or reducing our debt
as a percentage of GDP.

00:06:16.280 --> 00:06:19.620
This first chart
right over here is

00:06:19.620 --> 00:06:21.760
the-- we'll read
it right over here.

00:06:21.760 --> 00:06:26.122
It's the percent of our
population that is employed.

00:06:26.122 --> 00:06:27.580
And you see right--
so it's a proxy

00:06:27.580 --> 00:06:29.464
for employment or unemployment.

00:06:29.464 --> 00:06:30.880
Or, in this case,
it's employment.

00:06:30.880 --> 00:06:33.650
And you see we hit
a major recession

00:06:33.650 --> 00:06:35.984
at the end of end of 2008.

00:06:35.984 --> 00:06:37.650
You saw the percentage
of our population

00:06:37.650 --> 00:06:40.650
that is employed go
down dramatically.

00:06:40.650 --> 00:06:42.740
We kind of hit bottom
over here, and we're

00:06:42.740 --> 00:06:47.170
having a shaky-- hopefully,
a shaky recovery right now.

00:06:47.170 --> 00:06:49.760
It's not even clear
how good that will be.

00:06:49.760 --> 00:06:53.120
But hopefully we're having
a shaky recovery of sorts.

00:06:53.120 --> 00:06:55.920
And so just as we're
starting to recover,

00:06:55.920 --> 00:07:01.610
it might not be helpful to
take, essentially, $500 billion

00:07:01.610 --> 00:07:05.360
out of the economy in
order to pay down debt.

00:07:05.360 --> 00:07:07.420
So the fiscal cliff
scenario might

00:07:07.420 --> 00:07:11.770
make us go back into a
little bit of a lurch.

00:07:11.770 --> 00:07:14.850
And most economists
are predicting about

00:07:14.850 --> 00:07:18.450
on the order of 1% slower
growth if the fiscal cliff

00:07:18.450 --> 00:07:20.390
scenario were to hit.

00:07:20.390 --> 00:07:22.490
Actually, most
economists-- not 1%,

00:07:22.490 --> 00:07:26.610
they're actually expecting
about a 2% hit to the economy

00:07:26.610 --> 00:07:29.120
if the fiscal cliff
scenario would happen.

00:07:29.120 --> 00:07:32.570
And rather from going
from 1.7% growth,

00:07:32.570 --> 00:07:35.860
we might go to negative--
a slight recession,

00:07:35.860 --> 00:07:39.130
slight receding of the economy.

00:07:39.130 --> 00:07:42.990
So that's obviously not a good
scenario that we want to be in.

00:07:42.990 --> 00:07:45.370
The other chart that I
have right over here,

00:07:45.370 --> 00:07:48.800
these are yields on 10
year treasury notes.

00:07:48.800 --> 00:07:51.029
This seems like a very
technical thing to talk about.

00:07:51.029 --> 00:07:52.570
But this is essentially
the interest.

00:07:52.570 --> 00:07:53.320
One way to think
about it is this

00:07:53.320 --> 00:07:54.778
is the interest
that the government

00:07:54.778 --> 00:07:57.960
is paying when it borrows
money for 10 years.

00:07:57.960 --> 00:08:02.570
And one of the main arguments
for paying down debt

00:08:02.570 --> 00:08:05.510
is that, if the government
keeps borrowing money,

00:08:05.510 --> 00:08:07.890
it's going to crowd
out other borrowers.

00:08:07.890 --> 00:08:10.480
It's going to make
demand for money high.

00:08:10.480 --> 00:08:12.880
And maybe the supply will
be more and more scarce.

00:08:12.880 --> 00:08:15.400
And interest rates will go up.

00:08:15.400 --> 00:08:18.210
So a lot of borrowing could
make interest rates go up.

00:08:18.210 --> 00:08:20.470
And then the other
idea is, well,

00:08:20.470 --> 00:08:23.880
if there's a lot of just
borrowing and spending going

00:08:23.880 --> 00:08:24.950
on.

00:08:24.950 --> 00:08:27.410
And if we don't have a lot
of productive capacity,

00:08:27.410 --> 00:08:29.110
it could also lead to inflation.

00:08:29.110 --> 00:08:32.400
Or maybe when the
government has a huge debt,

00:08:32.400 --> 00:08:36.600
they have an incentive to try
to have inflationary policies.

00:08:36.600 --> 00:08:39.309
So that in real terms,
that debt seems to be less.

00:08:39.309 --> 00:08:41.679
But when you look
at the debt markets,

00:08:41.679 --> 00:08:44.030
and these are market
driven numbers.

00:08:44.030 --> 00:08:46.710
The Federal Reserve does
not control 10 year yields.

00:08:46.710 --> 00:08:51.110
The Federal Reserve
controls short term yields.

00:08:51.110 --> 00:08:53.630
You see that we have
historically low interest

00:08:53.630 --> 00:08:54.380
rates.

00:08:54.380 --> 00:08:58.600
So the bond market-- and
there's some very smart people

00:08:58.600 --> 00:09:02.650
who make their living
investing in the bond markets,

00:09:02.650 --> 00:09:04.830
one of the biggest markets,
if not the biggest.

00:09:04.830 --> 00:09:08.400
It is saying it's really
not worried about deficits.

00:09:08.400 --> 00:09:10.290
It's willing to lend
to the US government

00:09:10.290 --> 00:09:12.930
at all time low interest rates.

00:09:12.930 --> 00:09:14.500
It's not expecting
any inflation.

00:09:14.500 --> 00:09:16.500
If anything, the
bond market is saying

00:09:16.500 --> 00:09:18.000
that the biggest
risk in the economy

00:09:18.000 --> 00:09:19.830
is continued sluggish growth.

00:09:19.830 --> 00:09:22.010
The biggest risk in
the economy is maybe

00:09:22.010 --> 00:09:24.670
even some form of deflation.

00:09:24.670 --> 00:09:27.880
So this super low bond
yields right over here

00:09:27.880 --> 00:09:30.344
would be an argument
that well, yeah,

00:09:30.344 --> 00:09:32.260
if you believe what the
bond market is saying,

00:09:32.260 --> 00:09:35.710
we, one, definitely should not
suck money out of the economy.

00:09:35.710 --> 00:09:39.640
And the risks of maintaining
high levels of debt

00:09:39.640 --> 00:09:42.660
might not be as high as some
people are saying-- that you're

00:09:42.660 --> 00:09:44.880
not going to see interest
rates and inflation go

00:09:44.880 --> 00:09:47.280
through the roof tomorrow.

00:09:47.280 --> 00:09:51.150
Now, if you ascribe to the
second point of view that,

00:09:51.150 --> 00:09:52.630
OK, yes, we should
be responsible.

00:09:52.630 --> 00:09:56.000
We should start to at
minimum lower our deficit,

00:09:56.000 --> 00:10:00.550
and over time, as a percentage
of GDP, lower our debt.

00:10:00.550 --> 00:10:02.550
The other question is,
how do you do that?

00:10:02.550 --> 00:10:04.380
The Republicans would
want to do it mainly

00:10:04.380 --> 00:10:08.570
through spending cuts, while the
Democrats would want to do that

00:10:08.570 --> 00:10:14.177
possibly through spending
cuts, but also by letting some

00:10:14.177 --> 00:10:15.760
of the tax cuts on
the wealthy expire,

00:10:15.760 --> 00:10:17.800
and maybe even adding
some other tax.

00:10:17.800 --> 00:10:20.680
And so these two charts inform
some of those positions.

00:10:20.680 --> 00:10:23.540
And I'm trying to give as
balanced of an approach

00:10:23.540 --> 00:10:24.400
as I can.

00:10:24.400 --> 00:10:26.240
So for a Democrat,
they might look

00:10:26.240 --> 00:10:28.020
at this chart right over here.

00:10:28.020 --> 00:10:31.070
And they might say, well,
look, average tax rates

00:10:31.070 --> 00:10:32.880
for the highest
income taxpayers--

00:10:32.880 --> 00:10:34.930
if you look at a
historical basis,

00:10:34.930 --> 00:10:38.740
it looks like it's
relatively low.

00:10:38.740 --> 00:10:40.580
And a lot of this
comes out of the idea

00:10:40.580 --> 00:10:42.170
that even though
the marginal tax

00:10:42.170 --> 00:10:44.686
rates are higher than
what you see here,

00:10:44.686 --> 00:10:46.560
high income taxpayers--
a lot of their income

00:10:46.560 --> 00:10:48.870
might be from capital
gains, from dividends.

00:10:48.870 --> 00:10:51.630
Those are taxed at 15%.

00:10:51.630 --> 00:10:54.550
They also might have a more
sophisticated accountants

00:10:54.550 --> 00:10:56.140
who can get them
more deductions.

00:10:56.140 --> 00:10:58.059
And so that's why you
have them paying--

00:10:58.059 --> 00:11:00.350
and they're getting more
sophisticated as time goes by.

00:11:00.350 --> 00:11:04.310
And that's why you see
this trend going down.

00:11:04.310 --> 00:11:06.330
And so this could be an
argument, well at least

00:11:06.330 --> 00:11:10.880
relative to historical basis,
that this category of folks

00:11:10.880 --> 00:11:13.170
might be able to
afford to pay more.

00:11:13.170 --> 00:11:16.190
There's also another argument
in terms of a stimulus argument,

00:11:16.190 --> 00:11:18.065
or how to minimize the
impact on the economy.

00:11:21.400 --> 00:11:25.000
So if you have $1 that you
want to somehow stimulate

00:11:25.000 --> 00:11:27.180
the economy with.

00:11:27.180 --> 00:11:30.370
Well one, the government
could spend the money.

00:11:30.370 --> 00:11:32.030
And there are many
arguments-- very

00:11:32.030 --> 00:11:34.071
good arguments-- why the
government is not always

00:11:34.071 --> 00:11:35.650
the most efficient
spender of money,

00:11:35.650 --> 00:11:37.544
and not always spending
it in the best way.

00:11:37.544 --> 00:11:38.960
But they will
definitely spend it.

00:11:38.960 --> 00:11:40.560
So you give it to
the government.

00:11:40.560 --> 00:11:41.710
You give to the
government, they'll

00:11:41.710 --> 00:11:43.230
definitely spend
it, and then some.

00:11:43.230 --> 00:11:45.580
So they'll definitely
spend that dollar.

00:11:45.580 --> 00:11:48.440
At least it will enter
into the economy.

00:11:48.440 --> 00:11:53.150
Now, if you give it
to the middle class,

00:11:53.150 --> 00:11:54.670
they are also
likely to spend it.

00:11:54.670 --> 00:11:57.646
Or, to spend a good chunk of
it-- maybe save a little bit.

00:11:57.646 --> 00:11:59.520
And the argument would
be that if you give it

00:11:59.520 --> 00:12:06.690
to someone who's affluent,
they are likely to save it.

00:12:06.690 --> 00:12:09.620
Now, saving is not
always a bad thing.

00:12:09.620 --> 00:12:13.420
If you are low on investment,
you need more savers.

00:12:13.420 --> 00:12:17.410
If interest rates were
going through the roof

00:12:17.410 --> 00:12:18.170
right over here.

00:12:18.170 --> 00:12:20.890
That means that we don't
have enough supply of money.

00:12:20.890 --> 00:12:22.390
In which case, it
would make sense

00:12:22.390 --> 00:12:24.540
that we would try
to incent savers.

00:12:24.540 --> 00:12:27.330
So if you have a situation
of-- in the late '70s,

00:12:27.330 --> 00:12:29.660
where you have inflation
going through the roof.

00:12:29.660 --> 00:12:31.810
You want people to
actually invest more.

00:12:31.810 --> 00:12:34.310
You want a higher
supply of money.

00:12:34.310 --> 00:12:37.270
Makes complete sense
to put more money

00:12:37.270 --> 00:12:40.110
in the hands of people who
are likely to save and invest

00:12:40.110 --> 00:12:41.110
that money.

00:12:41.110 --> 00:12:43.230
But here, at least if you
believe the bond market,

00:12:43.230 --> 00:12:45.410
it looks like we have
a demand problem.

00:12:45.410 --> 00:12:47.880
That it's actually very
cheap to get capital.

00:12:47.880 --> 00:12:50.640
If anything, there's a
surplus of capital out there.

00:12:50.640 --> 00:12:52.620
But there's not
enough demand in order

00:12:52.620 --> 00:12:58.190
for people to get the economy
to run at its potential.

00:12:58.190 --> 00:12:59.950
So a Democratic
argument would be,

00:12:59.950 --> 00:13:04.320
look, if we're going
to have this dollar,

00:13:04.320 --> 00:13:06.040
the person that
we're best off giving

00:13:06.040 --> 00:13:08.550
it to is the middle class.

00:13:08.550 --> 00:13:10.620
Or possibly to some
degree, the government.

00:13:10.620 --> 00:13:12.120
Because at least
they will spend it.

00:13:12.120 --> 00:13:14.320
It will help stimulate
the economy here,

00:13:14.320 --> 00:13:16.310
it will go into
savings, but that

00:13:16.310 --> 00:13:19.590
doesn't seem like what the
government needs right now.

00:13:19.590 --> 00:13:22.090
Now, the counter argument that
you might get from someone

00:13:22.090 --> 00:13:24.830
at the right is that, first,
the government is just

00:13:24.830 --> 00:13:27.300
a hugely inefficient
spender of capital.

00:13:27.300 --> 00:13:29.630
And there's a lot of
evidence to-- you definitely

00:13:29.630 --> 00:13:31.796
need a government, but they
don't always spend money

00:13:31.796 --> 00:13:34.270
in the most efficient--
and even sometimes it

00:13:34.270 --> 00:13:36.400
leads to levels of corruption.

00:13:36.400 --> 00:13:38.360
And often times,
when you do something

00:13:38.360 --> 00:13:41.915
in the name of stimulus,
maybe it might make sense

00:13:41.915 --> 00:13:44.960
to do it in the short term, but
once that program is in place,

00:13:44.960 --> 00:13:47.250
it's very hard to
remove that program.

00:13:47.250 --> 00:13:50.267
People start to depend on it,
even when you, in theory, you

00:13:50.267 --> 00:13:51.600
don't need the stimulus anymore.

00:13:51.600 --> 00:13:56.140
It's a very hard to shrink
the size of government.

00:13:56.140 --> 00:13:58.450
And there's some
argument for that.

00:13:58.450 --> 00:14:00.730
If you look at outlays--
government expenditures

00:14:00.730 --> 00:14:03.230
as a percentage of
GDP-- it does look

00:14:03.230 --> 00:14:06.210
like, on a historical
basis, it is high.

00:14:06.210 --> 00:14:09.600
Now, to some degree,
this is due to the fact

00:14:09.600 --> 00:14:12.750
that we have entered into kind
of a major recessionary phase.

00:14:12.750 --> 00:14:16.460
You see that in all of
the major recessions,

00:14:16.460 --> 00:14:22.540
the government gets less revenue
and has to do more outlays.

00:14:22.540 --> 00:14:27.320
This happens in every-- well,
I would say in most recessions.

00:14:27.320 --> 00:14:29.210
And this happened
in dramatic form,

00:14:29.210 --> 00:14:32.450
where we started to
have more outlays.

00:14:32.450 --> 00:14:35.720
We had-- actually this point, we
had wars that we were funding.

00:14:35.720 --> 00:14:40.010
And we were doing these trillion
dollar stimulus packages.

00:14:40.010 --> 00:14:41.690
And we have these
trillion bailouts.

00:14:41.690 --> 00:14:44.980
At the same time
that the economy

00:14:44.980 --> 00:14:46.450
was receding right over here.

00:14:46.450 --> 00:14:47.920
But just when you
look at this, it

00:14:47.920 --> 00:14:50.150
does look like the
government spending,

00:14:50.150 --> 00:14:53.910
as a percentage of GDP,
is at historical highs.

00:14:53.910 --> 00:14:56.170
And as we say, as the
arguement would have it,

00:14:56.170 --> 00:14:57.740
it's very hard to
get that to shrink,

00:14:57.740 --> 00:14:59.310
even when it should
be shrinking.

00:14:59.310 --> 00:15:05.660
And it's often being deployed
in less than efficient ways.

00:15:05.660 --> 00:15:07.620
So I will leave you there.

00:15:07.620 --> 00:15:10.610
My intent really is not to
sway you one way or the other.

00:15:10.610 --> 00:15:13.660
But to really just hope that
you have good information when

00:15:13.660 --> 00:15:18.544
you think about the issues
surrounding the fiscal cliff.


The fiscal cliff

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.150
There's been a lot of talk
lately about the fiscal cliff,

00:00:04.150 --> 00:00:06.262
which sounds very dramatic.

00:00:06.262 --> 00:00:07.720
And what I want to
do in this video

00:00:07.720 --> 00:00:09.370
is at least lay
out the numbers so

00:00:09.370 --> 00:00:10.920
that we're all on the same page.

00:00:10.920 --> 00:00:14.380
And then the next few videos,
think about the implications

00:00:14.380 --> 00:00:17.310
or how the negotiations between
the president and Congress

00:00:17.310 --> 00:00:19.210
might actually work out.

00:00:19.210 --> 00:00:22.930
So the first thing to
frame this conversation

00:00:22.930 --> 00:00:25.150
is just where the
budget is right now.

00:00:25.150 --> 00:00:27.590
And so let me draw
a little graph.

00:00:27.590 --> 00:00:33.570
So let me make this axis right
over here represent our budget.

00:00:33.570 --> 00:00:35.090
I want to make it
as big as possible

00:00:35.090 --> 00:00:37.270
so that we can get a little
bit of granular detail

00:00:37.270 --> 00:00:40.080
on how much the budget
might move based

00:00:40.080 --> 00:00:42.620
on how the
negotiations work out.

00:00:42.620 --> 00:00:47.010
So let's say that this line
is roughly $4 trillion long.

00:00:47.010 --> 00:00:49.780
So that's $4 trillion,
and halfway would

00:00:49.780 --> 00:00:51.970
be about $2 trillion.

00:00:51.970 --> 00:00:55.060
And then in between two and
four, you've got $3 trillion.

00:00:55.060 --> 00:01:00.045
And then $1 trillion would be
right over here, $1 trillion.

00:01:00.045 --> 00:01:01.420
And then I'm going
to do some bar

00:01:01.420 --> 00:01:04.640
charts to show the
different scenarios.

00:01:04.640 --> 00:01:08.470
And as I do this, keep in mind
the size of the US economy.

00:01:08.470 --> 00:01:13.520
The US economy is
approximately $15.5 trillion.

00:01:13.520 --> 00:01:17.970
So that is roughly our
GDP, depends on which year

00:01:17.970 --> 00:01:19.380
you're measuring.

00:01:19.380 --> 00:01:21.840
But that gives us at
least a frame of reference

00:01:21.840 --> 00:01:24.420
for what chunk of
GDP we are talking

00:01:24.420 --> 00:01:29.480
about when we talk about
the federal budget.

00:01:29.480 --> 00:01:33.780
So let's start with the 2012.

00:01:33.780 --> 00:01:36.360
Let's start with
the 2012 budget.

00:01:36.360 --> 00:01:43.130
So in 2012, the US government
is spending $3.6 trillion.

00:01:43.130 --> 00:01:45.950
So let me make my graph a
little bit more granular.

00:01:45.950 --> 00:01:48.170
So this would be 3.5.

00:01:48.170 --> 00:01:51.600
So 3.6 is going to
be right around here.

00:01:51.600 --> 00:01:54.040
So let me draw that.

00:01:54.040 --> 00:01:57.670
And I'll do it in this purple
color for the expenditures.

00:01:57.670 --> 00:02:02.880
So this is how much the
federal government spent

00:02:02.880 --> 00:02:06.500
or I guess is spending in 2012.

00:02:06.500 --> 00:02:09.539
So just like that.

00:02:09.539 --> 00:02:11.430
All right, there we go.

00:02:11.430 --> 00:02:13.290
So that's expenditures.

00:02:13.290 --> 00:02:15.810
Now you are probably
aware that we

00:02:15.810 --> 00:02:17.280
don't have all of the revenue.

00:02:17.280 --> 00:02:20.770
We didn't bring in the
$3.6 trillion in taxes.

00:02:20.770 --> 00:02:24.140
So this right over
here is $3.6 trillion.

00:02:24.140 --> 00:02:27.090
Our revenue that we get through
tax revenue and other things

00:02:27.090 --> 00:02:29.010
is somewhat less.

00:02:29.010 --> 00:02:36.300
It, in 2012, was on the
order of $2.5 trillion.

00:02:36.300 --> 00:02:38.150
So I'll draw that
right over here.

00:02:38.150 --> 00:02:39.680
So $2.5 trillion.

00:02:39.680 --> 00:02:41.900
I'll do it in this green color.

00:02:41.900 --> 00:02:46.910
So this is how much
revenue was brought in.

00:02:46.910 --> 00:02:49.340
So let me write this down.

00:02:49.340 --> 00:02:54.280
This is $2.5 trillion.

00:02:54.280 --> 00:02:56.920
Now let's think
about how much might

00:02:56.920 --> 00:03:00.350
get spent under the
different scenarios.

00:03:00.350 --> 00:03:04.270
So first I'll lay out a rough
approximation of Obama's budget

00:03:04.270 --> 00:03:06.770
proposal for 2013.

00:03:06.770 --> 00:03:11.390
So Obama in 2013.

00:03:11.390 --> 00:03:15.590
So on the spending side, he
sees, or he would like to see,

00:03:15.590 --> 00:03:19.700
spending go up by $200 billion.

00:03:19.700 --> 00:03:22.040
So let's see this.

00:03:22.040 --> 00:03:26.315
So on the spending side, we're
going to add $200 billion.

00:03:26.315 --> 00:03:27.690
And the next few
videos, we could

00:03:27.690 --> 00:03:29.880
talk about the pros
and cons, the arguments

00:03:29.880 --> 00:03:32.690
for and against
something like that.

00:03:32.690 --> 00:03:34.670
So let me draw that.

00:03:34.670 --> 00:03:46.280
So relative to the 3.6,
we're now that 3.8.

00:03:46.280 --> 00:03:53.090
So you have a spending increase
of-- so plus $200 billion.

00:03:53.090 --> 00:03:59.140
That gets us to 3.8 billion
in total expenditures.

00:03:59.140 --> 00:04:03.080
If my best estimate of what
the Republicans in Congress

00:04:03.080 --> 00:04:04.020
would want.

00:04:04.020 --> 00:04:05.150
So let's write.

00:04:05.150 --> 00:04:10.130
Let's say, a Republicans in
2013 is that they would actually

00:04:10.130 --> 00:04:13.730
ideally want spending
cuts from these levels.

00:04:13.730 --> 00:04:15.190
So let's draw that out.

00:04:15.190 --> 00:04:18.860
And roughly on the order
of about $100 billion.

00:04:18.860 --> 00:04:20.527
So roughly on the
order of $100 billion.

00:04:20.527 --> 00:04:22.026
They might even
want more than that,

00:04:22.026 --> 00:04:23.550
but let's just go
with that for now.

00:04:23.550 --> 00:04:28.720
So that gets us to a
budget of $3.5 trillion.

00:04:28.720 --> 00:04:30.680
That's about that
right over there.

00:04:33.850 --> 00:04:37.900
$3.5 trillion.

00:04:37.900 --> 00:04:42.960
So once again, relative
to 2012, you're

00:04:42.960 --> 00:04:45.070
going down by $100 billion.

00:04:45.070 --> 00:04:46.820
Let me make clear.

00:04:46.820 --> 00:04:50.020
You're subtracting $100 billion.

00:04:50.020 --> 00:04:55.290
Now in the fiscal
cliff scenario,

00:04:55.290 --> 00:04:58.660
the spending will be similar
to the Republican ideal right

00:04:58.660 --> 00:04:59.230
over here.

00:04:59.230 --> 00:05:00.521
So let me write this over here.

00:05:00.521 --> 00:05:01.350
Fiscal cliff.

00:05:04.770 --> 00:05:06.020
We are also spending.

00:05:06.020 --> 00:05:08.820
We are cutting on the
order of $100 billion

00:05:08.820 --> 00:05:11.200
in government expenditures.

00:05:11.200 --> 00:05:13.070
So let's draw that.

00:05:13.070 --> 00:05:14.900
So at least on the
expenditure side.

00:05:14.900 --> 00:05:16.440
And these are all very rough.

00:05:16.440 --> 00:05:18.830
I'm sure the Republicans
who would agree and disagree

00:05:18.830 --> 00:05:19.330
with this.

00:05:19.330 --> 00:05:23.980
But I'm trying to get
my best sense of kind

00:05:23.980 --> 00:05:27.050
of an aggregate view on things.

00:05:27.050 --> 00:05:29.560
So the fiscal cliff.

00:05:29.560 --> 00:05:33.220
We are also cutting
spending by $100 billion.

00:05:39.950 --> 00:05:43.770
Now let's go to the
revenue side of things.

00:05:43.770 --> 00:05:47.890
In all of these scenarios for
2013, and just to be clear,

00:05:47.890 --> 00:05:50.280
the fiscal cliff
that's also for 2013.

00:05:50.280 --> 00:05:54.570
And all of these scenarios,
we get the same revenue

00:05:54.570 --> 00:05:56.370
that we got in 2012.

00:05:56.370 --> 00:05:58.220
So let me draw that.

00:05:58.220 --> 00:06:00.590
Plus we get about
another $100 billion

00:06:00.590 --> 00:06:02.400
from the growth in the economy.

00:06:02.400 --> 00:06:04.700
As the economy grows, and
even if your tax rates are

00:06:04.700 --> 00:06:06.360
held completely
constant, you're going

00:06:06.360 --> 00:06:10.600
to get more revenue for
the federal government.

00:06:10.600 --> 00:06:14.160
You get about $100 billion
from the federal government.

00:06:14.160 --> 00:06:18.190
And so that gets
us to $2.6 trillion

00:06:18.190 --> 00:06:21.170
without changing anything.

00:06:21.170 --> 00:06:23.300
So let me just shade all
of these in really fast.

00:06:27.920 --> 00:06:30.570
So shade that one in.

00:06:30.570 --> 00:06:32.440
Shade that one in.

00:06:32.440 --> 00:06:36.490
And then, shade that one in.

00:06:36.490 --> 00:06:40.600
Now, as you've probably
heard on the news,

00:06:40.600 --> 00:06:44.790
Obama would like to
extend the Bush tax

00:06:44.790 --> 00:06:45.960
cuts for the middle class.

00:06:45.960 --> 00:06:48.330
And he considers a
middle class of those

00:06:48.330 --> 00:06:53.140
who are making less than
$250,000 for a family.

00:06:53.140 --> 00:06:58.440
But he would like to not extend
the Bush tax cuts on the rich.

00:06:58.440 --> 00:07:02.330
And he would like to actually
include a few other tax

00:07:02.330 --> 00:07:04.850
increases, also on the wealthy.

00:07:04.850 --> 00:07:07.810
And so you would get
an increase of revenue

00:07:07.810 --> 00:07:11.170
under Obama's plan
of $300 billion.

00:07:13.750 --> 00:07:16.680
This is once again, very rough.

00:07:16.680 --> 00:07:19.750
Probably $50, $60 billion that
I'm not fully accounting for,

00:07:19.750 --> 00:07:21.710
but it'll give you
the rough picture.

00:07:21.710 --> 00:07:24.020
So this is $300 billion.

00:07:24.020 --> 00:07:27.790
And what Obama's doing here, or
what at least in the proposal,

00:07:27.790 --> 00:07:31.200
as far as I can believe,
none of this is that simple.

00:07:31.200 --> 00:07:33.240
What they're talking
about right over here

00:07:33.240 --> 00:07:44.250
is extend tax cuts for middle
class, which my best reading,

00:07:44.250 --> 00:07:48.120
seems like we would lose a
little under $200 billion

00:07:48.120 --> 00:07:49.270
of revenue.

00:07:49.270 --> 00:07:53.280
But then we keep the tax
cuts or we let the tax cuts

00:07:53.280 --> 00:07:56.140
on the wealthy expire.

00:07:56.140 --> 00:07:58.050
So that gets us 200 of this.

00:07:58.050 --> 00:07:59.940
And then there are
other tax increases

00:07:59.940 --> 00:08:03.230
and another removing
loopholes and whatever

00:08:03.230 --> 00:08:06.700
else they increase
this to $300 billion.

00:08:06.700 --> 00:08:09.470
So let's compare the deficit.

00:08:09.470 --> 00:08:12.130
So this gets us to,
in the Obama scenario,

00:08:12.130 --> 00:08:16.020
we end up with $2.9
trillion in revenue.

00:08:16.020 --> 00:08:17.970
So let's compare
what the deficit did

00:08:17.970 --> 00:08:21.180
from 2012 to
Obama's budget plan.

00:08:21.180 --> 00:08:25.590
So in 2012, if you
take $3.6 trillion,

00:08:25.590 --> 00:08:32.409
subtract out $2.5 trillion,
there is a gap $1.1 trillion.

00:08:32.409 --> 00:08:33.850
This is the deficit.

00:08:33.850 --> 00:08:38.230
This is how much the government
has to borrow in 2012.

00:08:38.230 --> 00:08:41.169
Under Obama's budget,
what would it be for 2013?

00:08:41.169 --> 00:08:44.049
Well we're spending
$3.8 trillion.

00:08:44.049 --> 00:08:47.040
We are getting a $2.9 trillion.

00:08:47.040 --> 00:08:51.165
So you have a gap
of $900 billion.

00:08:54.150 --> 00:08:57.060
So there is some of
the deficit reduction,

00:08:57.060 --> 00:08:59.510
although the deficit
is still quite large.

00:08:59.510 --> 00:09:01.780
The deficit reduction
is $200 billion.

00:09:01.780 --> 00:09:05.430
$100 billion of that came
from the economic growth.

00:09:05.430 --> 00:09:10.580
And then, the rest is coming
from, or a good chunk of that,

00:09:10.580 --> 00:09:13.396
is coming from increased taxes.

00:09:13.396 --> 00:09:15.520
Depending on how you view
it, either increase taxes

00:09:15.520 --> 00:09:18.600
on the wealthy or not
letting the tax cuts

00:09:18.600 --> 00:09:20.850
expire on the wealthy.

00:09:20.850 --> 00:09:23.690
Now let's think about
the Republican situation.

00:09:23.690 --> 00:09:26.300
Well, you have $2.6
trillion in revenue.

00:09:26.300 --> 00:09:29.040
And you have $3.5 trillion.

00:09:29.040 --> 00:09:30.190
Let me write this down.

00:09:35.470 --> 00:09:44.420
And once again, you have on the
order of a $900 billion gap.

00:09:44.420 --> 00:09:46.540
So in terms of
deficit reduction,

00:09:46.540 --> 00:09:48.230
these things look
pretty similar.

00:09:48.230 --> 00:09:50.570
You have a very similar deficit.

00:09:50.570 --> 00:09:52.507
Obama is increasing spending.

00:09:52.507 --> 00:09:54.090
And he would argue
that he's investing

00:09:54.090 --> 00:09:56.110
in things that might help
stimulate the economy,

00:09:56.110 --> 00:09:58.270
or invest in America
for the future.

00:09:58.270 --> 00:10:00.810
And then he's making it
up by letting the tax cuts

00:10:00.810 --> 00:10:03.480
on the wealthy expire
for the most part.

00:10:03.480 --> 00:10:05.820
The Republicans want
to cut spending.

00:10:05.820 --> 00:10:08.520
But they're also letting
the tax cuts continue.

00:10:08.520 --> 00:10:13.410
So you essentially have the
same level of deficit reduction.

00:10:13.410 --> 00:10:17.980
Now I think we are ready to
talk about the fiscal cliff.

00:10:17.980 --> 00:10:21.910
The fiscal cliff, we're
spending hundreds billion less.

00:10:21.910 --> 00:10:25.450
And then we are also letting
all of the tax cuts for both

00:10:25.450 --> 00:10:27.500
the wealthy, those
who are earning

00:10:27.500 --> 00:10:30.170
at a family level
more than 250,000.

00:10:30.170 --> 00:10:33.800
And for the middle class,
we're letting them all expire.

00:10:33.800 --> 00:10:39.170
And so you have the revenue
increase by $400 billion.

00:10:39.170 --> 00:10:41.290
So this goes up by $400 billion.

00:10:45.080 --> 00:10:46.980
And so that takes us
roughly, once again,

00:10:46.980 --> 00:10:49.080
this is all rough,
to about $3 trillion

00:10:49.080 --> 00:10:53.610
in revenue and $3.5
trillion in expenses.

00:10:53.610 --> 00:10:57.040
And so your deficit, under
the fiscal cliff scenario,

00:10:57.040 --> 00:11:02.090
the deficit is going
to be $500 billion.

00:11:02.090 --> 00:11:03.880
Now you might say,
hey, this is great.

00:11:03.880 --> 00:11:05.480
Everyone talks
about the deficit.

00:11:05.480 --> 00:11:07.050
The deficit is a scary thing.

00:11:07.050 --> 00:11:09.890
We are borrowing from
the future and all that.

00:11:09.890 --> 00:11:13.030
Why are people so afraid
of the fiscal cliff?

00:11:13.030 --> 00:11:15.770
The reality is that if
you take $500 billion

00:11:15.770 --> 00:11:16.730
out of the economy.

00:11:16.730 --> 00:11:22.660
So $100 billion through spending
cuts and then $400 billion

00:11:22.660 --> 00:11:24.550
from tax increases.

00:11:24.550 --> 00:11:25.990
So the government
is deleveraging.

00:11:25.990 --> 00:11:28.031
But that money's being
sucked out of the economy.

00:11:28.031 --> 00:11:30.530
And you could argue that there's
just kind of a multiplier

00:11:30.530 --> 00:11:34.120
effect as well, that might
endanger what's already

00:11:34.120 --> 00:11:36.260
a very precarious
risk of recovery.

00:11:36.260 --> 00:11:40.690
That the recovery is really
just starting to happen.

00:11:40.690 --> 00:11:43.680
And if we were to suck all of
this money out of the economy,

00:11:43.680 --> 00:11:45.960
that's what the
argument would be,

00:11:45.960 --> 00:11:48.100
then that might throw us
into another recession.

00:11:48.100 --> 00:11:51.340
Or that might make the
recovery that much weaker.

00:11:51.340 --> 00:11:53.117
In the next few
videos, we'll discuss

00:11:53.117 --> 00:11:54.450
that in a little bit more depth.

00:11:54.450 --> 00:11:57.460
See what people are saying
the impact might be.

00:11:57.460 --> 00:12:00.734
And what the arguments
might be in either case.


Another Pythagorean theorem proof

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.170
In case you haven't
noticed, I've

00:00:02.170 --> 00:00:04.340
gotten somewhat
obsessed with doing

00:00:04.340 --> 00:00:06.460
as many proofs of the
Pythagorean theorem

00:00:06.460 --> 00:00:07.610
as I can do.

00:00:07.610 --> 00:00:09.220
So let's do one more.

00:00:09.220 --> 00:00:11.070
And like how all of
these proofs start,

00:00:11.070 --> 00:00:14.750
let's construct ourselves
a right triangle.

00:00:14.750 --> 00:00:18.120
So I'm going to construct it
so that its hypotenuse sits

00:00:18.120 --> 00:00:19.600
on the bottom.

00:00:19.600 --> 00:00:22.500
So that's the hypotenuse
of my right triangle.

00:00:22.500 --> 00:00:23.940
Try to draw it as
big as possible,

00:00:23.940 --> 00:00:26.260
so that we have
space to work with.

00:00:26.260 --> 00:00:28.470
So that's going to
be my hypotenuse.

00:00:28.470 --> 00:00:32.680
And then let's say
that this is the longer

00:00:32.680 --> 00:00:34.390
side that's not the hypotenuse.

00:00:34.390 --> 00:00:36.260
We can have two
sides that are equal.

00:00:36.260 --> 00:00:38.770
But I'll just draw it so that
it looks a little bit longer.

00:00:38.770 --> 00:00:41.010
Let's call that side length a.

00:00:41.010 --> 00:00:44.080
And then let's draw this
side right over here.

00:00:44.080 --> 00:00:45.620
It has to be a right triangle.

00:00:45.620 --> 00:00:48.110
So maybe it goes
right over there.

00:00:48.110 --> 00:00:49.960
That's side of length b.

00:00:49.960 --> 00:00:52.560
Let me extend the
length a a little bit.

00:00:52.560 --> 00:00:56.090
So it definitely looks
like a right triangle.

00:00:56.090 --> 00:01:00.020
And this is our 90-degree angle.

00:01:00.020 --> 00:01:02.850
So the first thing
that I'm going to do

00:01:02.850 --> 00:01:06.170
is take this triangle
and then rotate

00:01:06.170 --> 00:01:09.310
it counterclockwise
by 90 degrees.

00:01:09.310 --> 00:01:12.100
So if I rotate it
counterclockwise by 90 degrees,

00:01:12.100 --> 00:01:14.270
I'm literally just going
to rotate it like that

00:01:14.270 --> 00:01:17.840
and draw another completely
congruent version of this one.

00:01:17.840 --> 00:01:20.120
So I'm going to rotate
it by 90 degrees.

00:01:20.120 --> 00:01:22.580
And if I did that,
the hypotenuse

00:01:22.580 --> 00:01:24.920
would then sit straight up.

00:01:24.920 --> 00:01:27.920
So I'm going to do my best
attempt to draw it almost

00:01:27.920 --> 00:01:30.800
to scale as much as
I can eyeball it.

00:01:30.800 --> 00:01:38.160
This side of length a will
now look something like this.

00:01:38.160 --> 00:01:41.080
It'll actually be parallel
to this over here.

00:01:41.080 --> 00:01:44.670
So let me see how
well I can draw it.

00:01:44.670 --> 00:01:46.710
So this is the side of length a.

00:01:46.710 --> 00:01:49.870
And if we cared, this
would be 90 degrees.

00:01:49.870 --> 00:01:51.880
The rotation between
the corresponding sides

00:01:51.880 --> 00:01:53.796
are just going to be 90
degrees in every case.

00:01:53.796 --> 00:01:55.070
That's going to be 90 degrees.

00:01:55.070 --> 00:01:57.190
That's going to be 90 degrees.

00:01:57.190 --> 00:01:59.420
Now, let me draw side b.

00:01:59.420 --> 00:02:04.630
So it's going to look
something like that

00:02:04.630 --> 00:02:07.410
or the side that's length b.

00:02:07.410 --> 00:02:11.410
And this and the right
angle is now here.

00:02:11.410 --> 00:02:17.110
So all I did is I rotated this
by 90 degrees counterclockwise.

00:02:17.110 --> 00:02:19.980
Now, what I want to do is
construct a parallelogram.

00:02:19.980 --> 00:02:23.280
I'm going to construct a
parallelogram by essentially--

00:02:23.280 --> 00:02:24.440
and let me label.

00:02:24.440 --> 00:02:26.310
So this is height
c right over here.

00:02:26.310 --> 00:02:28.820
Let me do that white color.

00:02:28.820 --> 00:02:31.480
This is height c.

00:02:31.480 --> 00:02:33.560
Now, what I want to do
is go from this point

00:02:33.560 --> 00:02:35.210
and go up c as well.

00:02:41.910 --> 00:02:45.070
Now, so this is
height c as well.

00:02:45.070 --> 00:02:47.030
And what is this length?

00:02:47.030 --> 00:02:49.930
What is the length over here
from this point to this point

00:02:49.930 --> 00:02:50.500
going to be?

00:02:55.280 --> 00:02:58.900
Well, a little clue is
this is a parallelogram.

00:02:58.900 --> 00:03:00.360
This line right
over here is going

00:03:00.360 --> 00:03:01.850
to be parallel to this line.

00:03:01.850 --> 00:03:04.460
It's maintained
the same distance.

00:03:04.460 --> 00:03:06.830
And since it's traveling
the same distance

00:03:06.830 --> 00:03:09.690
in the x direction or in
the horizontal direction

00:03:09.690 --> 00:03:12.460
and the vertical direction, this
is going to be the same length.

00:03:12.460 --> 00:03:16.110
So this is going
to be of length a.

00:03:16.110 --> 00:03:18.880
Now, the next question
I have for you

00:03:18.880 --> 00:03:23.410
is, what is the area of this
parallelogram that I have just

00:03:23.410 --> 00:03:26.180
constructed?

00:03:26.180 --> 00:03:29.320
Well, to think about that, let's
redraw this part of the diagram

00:03:29.320 --> 00:03:32.500
so that the parallelogram
is sitting on the ground.

00:03:32.500 --> 00:03:34.240
So this is length a.

00:03:39.410 --> 00:03:42.530
This is length c.

00:03:42.530 --> 00:03:44.540
This is length c.

00:03:44.540 --> 00:03:47.010
And if you look at this
part right over here,

00:03:47.010 --> 00:03:47.980
it gives you a clue.

00:03:50.267 --> 00:03:51.350
I'll use this green color.

00:03:51.350 --> 00:03:54.570
The height of the parallelogram
is given right over here.

00:03:54.570 --> 00:03:58.920
This side is
perpendicular to the base.

00:03:58.920 --> 00:04:05.900
So the height of the
parallelogram is a as well.

00:04:05.900 --> 00:04:07.430
So what's the area?

00:04:07.430 --> 00:04:09.230
Well, the area of
a parallelogram

00:04:09.230 --> 00:04:12.320
is just the base
times the height.

00:04:12.320 --> 00:04:15.070
So the area of this
parallelogram right over here

00:04:15.070 --> 00:04:16.749
is going to be a squared.

00:04:19.342 --> 00:04:20.550
Now, let's do the same thing.

00:04:20.550 --> 00:04:23.050
But let's rotate our
original right triangle.

00:04:23.050 --> 00:04:25.430
Let's rotate it the other way.

00:04:25.430 --> 00:04:28.380
So let's rotate it
90 degrees clockwise.

00:04:28.380 --> 00:04:30.850
And this time, instead of
pivoting on this point,

00:04:30.850 --> 00:04:34.530
we're going to pivot on
that point right over there.

00:04:34.530 --> 00:04:36.050
So what are we going to get?

00:04:36.050 --> 00:04:39.750
So the side of length
c if we rotate it

00:04:39.750 --> 00:04:46.217
like that, it's going to
end up right over here.

00:04:46.217 --> 00:04:48.300
I'll try to draw it as
close to scale as possible.

00:04:48.300 --> 00:04:51.350
So that side has length c.

00:04:51.350 --> 00:04:53.760
Now, the side of
length of b is going

00:04:53.760 --> 00:04:57.884
to pop out and look
something like this.

00:04:57.884 --> 00:04:59.300
It's going to be
parallel to that.

00:04:59.300 --> 00:05:01.130
This is going to
be a right angle.

00:05:01.130 --> 00:05:04.310
So let me draw it like that.

00:05:04.310 --> 00:05:05.980
That looks pretty good.

00:05:05.980 --> 00:05:09.920
And then the side of length
a is going to be out here.

00:05:14.370 --> 00:05:15.880
So that's a.

00:05:15.880 --> 00:05:18.460
And then this right
over here is b.

00:05:18.460 --> 00:05:21.590
And I wanted to
do that b in blue.

00:05:21.590 --> 00:05:24.790
So let me do the b in blue.

00:05:24.790 --> 00:05:27.710
And then this right
angle once we've rotated

00:05:27.710 --> 00:05:30.770
is just sitting right over here.

00:05:30.770 --> 00:05:32.250
Now, let's do the same exercise.

00:05:32.250 --> 00:05:35.910
Let's construct a
parallelogram right over here.

00:05:35.910 --> 00:05:37.420
So this is height c.

00:05:37.420 --> 00:05:39.570
This is height c as well.

00:05:39.570 --> 00:05:43.350
So by the same logic we used
over here, if this length is b,

00:05:43.350 --> 00:05:44.830
this length is b as well.

00:05:44.830 --> 00:05:46.300
These are parallel lines.

00:05:46.300 --> 00:05:49.950
We're going the same distance
in the horizontal direction.

00:05:49.950 --> 00:05:52.080
We're rising the same in
the vertical direction.

00:05:52.080 --> 00:05:53.920
We know that because
they're parallel.

00:05:53.920 --> 00:05:55.360
So this is length b down here.

00:05:55.360 --> 00:05:57.390
This is length b up there.

00:05:57.390 --> 00:06:01.890
Now, what is the area of this
parallelogram right over there?

00:06:01.890 --> 00:06:05.870
What is the area of that
parallelogram going to be?

00:06:05.870 --> 00:06:08.910
Well, once again to
help us visualize it,

00:06:08.910 --> 00:06:12.840
we can draw it sitting flat.

00:06:12.840 --> 00:06:14.470
So this is that side.

00:06:14.470 --> 00:06:16.980
Then you have another
side right over here.

00:06:16.980 --> 00:06:19.770
They both have length b.

00:06:19.770 --> 00:06:24.560
And you have the
sides of length c.

00:06:24.560 --> 00:06:25.480
So that's c.

00:06:25.480 --> 00:06:26.460
That's c.

00:06:26.460 --> 00:06:27.510
What is its height?

00:06:27.510 --> 00:06:29.400
Well, you see it
right over here.

00:06:29.400 --> 00:06:34.220
Its height is length b as well.

00:06:34.220 --> 00:06:35.220
It gives us right there.

00:06:35.220 --> 00:06:41.490
We know that this is 90 degrees.

00:06:41.490 --> 00:06:43.450
We did a 90-degree rotation.

00:06:43.450 --> 00:06:45.590
So this is how we
constructed the thing.

00:06:45.590 --> 00:06:47.960
So given that, the
area of a parallelogram

00:06:47.960 --> 00:06:50.690
is just the base
times the height.

00:06:50.690 --> 00:06:54.670
The area of this
parallelogram is b squared.

00:06:54.670 --> 00:07:00.150
So now, things are starting
to get interesting.

00:07:00.150 --> 00:07:04.420
And what I'm going to do is
I'm going to copy and paste

00:07:04.420 --> 00:07:09.200
this part right over here,
because this is, in my mind,

00:07:09.200 --> 00:07:11.830
the most interesting
part of our diagram.

00:07:11.830 --> 00:07:15.530
Let me see how well
I can select it.

00:07:15.530 --> 00:07:19.360
So let me select this
part right over here.

00:07:19.360 --> 00:07:21.120
So let me copy.

00:07:21.120 --> 00:07:23.980
And then I'm going
to scroll down.

00:07:23.980 --> 00:07:25.140
And then let me paste it.

00:07:27.900 --> 00:07:32.230
So this diagram that we've
constructed right over here,

00:07:32.230 --> 00:07:35.990
it's pretty clear what the area
of it is, the combined diagram.

00:07:35.990 --> 00:07:39.690
And let me delete
a few parts of it.

00:07:39.690 --> 00:07:42.450
I want to do that in black
so that it cleans it up.

00:07:42.450 --> 00:07:45.090
So let me clean this
thing up, so we really

00:07:45.090 --> 00:07:47.490
get the part that
we want to focus on.

00:07:47.490 --> 00:07:56.070
So cleaning that up and cleaning
this up, cleaning this up right

00:07:56.070 --> 00:07:58.760
over there.

00:07:58.760 --> 00:08:04.470
And actually, let me delete
this right down here as well,

00:08:04.470 --> 00:08:06.850
although we know that
this length was c.

00:08:06.850 --> 00:08:08.734
And actually, I'll draw
it right over here.

00:08:08.734 --> 00:08:10.400
This was from our
original construction.

00:08:10.400 --> 00:08:13.490
We know that this length is c.

00:08:13.490 --> 00:08:14.650
We know this height is c.

00:08:14.650 --> 00:08:16.600
We know this down here is c.

00:08:16.600 --> 00:08:19.780
But my question
for you is, what is

00:08:19.780 --> 00:08:25.420
the area of this combined shape?

00:08:25.420 --> 00:08:28.090
Well, it's just a
squared plus b squared.

00:08:28.090 --> 00:08:29.180
Let me write that down.

00:08:29.180 --> 00:08:33.270
The area is just a
squared plus b squared,

00:08:33.270 --> 00:08:36.530
the area of those
two parallelograms.

00:08:36.530 --> 00:08:42.070
Now, how can we maybe
rearrange pieces of this shape

00:08:42.070 --> 00:08:46.070
so that we can express
it in terms of c?

00:08:46.070 --> 00:08:47.790
Well, it might have
jumped out at you

00:08:47.790 --> 00:08:53.890
when I drew this
line right over here.

00:08:53.890 --> 00:08:55.900
I want to do that in white.

00:08:55.900 --> 00:09:01.230
We know that this part right
over here is of length c.

00:09:01.230 --> 00:09:03.456
This comes from our
original construction.

00:09:03.456 --> 00:09:03.955
Whoops.

00:09:03.955 --> 00:09:05.660
I lost my diagram.

00:09:05.660 --> 00:09:06.510
This is of length c.

00:09:06.510 --> 00:09:07.310
That's of length c.

00:09:07.310 --> 00:09:10.360
And then this right over
here is of length c.

00:09:10.360 --> 00:09:14.250
And so what we could do is take
this top right triangle, which

00:09:14.250 --> 00:09:16.960
is completely congruent to
our original right triangle,

00:09:16.960 --> 00:09:18.420
and shift it down.

00:09:18.420 --> 00:09:20.820
So remember, the entire area,
including this top right

00:09:20.820 --> 00:09:23.310
triangle, is a squared
plus b squared.

00:09:23.310 --> 00:09:25.257
But we're excluding
this part down here,

00:09:25.257 --> 00:09:26.590
which was our original triangle.

00:09:26.590 --> 00:09:29.620
But what happens
if we take that?

00:09:29.620 --> 00:09:33.790
So let me actually cut.

00:09:33.790 --> 00:09:34.900
And then let me paste it.

00:09:34.900 --> 00:09:38.480
And all I'm doing is I'm
moving that triangle down here.

00:09:38.480 --> 00:09:40.010
So now, it looks like this.

00:09:40.010 --> 00:09:44.470
So I've just rearranged the area
that was a squared b squared.

00:09:44.470 --> 00:09:47.300
So this entire area
of this entire square

00:09:47.300 --> 00:09:49.750
is still a squared
plus b squared.

00:09:49.750 --> 00:09:53.880
a squared is this entire
area right over here.

00:09:53.880 --> 00:09:55.350
It was before a parallelogram.

00:09:55.350 --> 00:09:58.530
I just shifted that top part
of the parallelogram down.

00:09:58.530 --> 00:10:04.302
b squared is this entire
area right over here.

00:10:04.302 --> 00:10:06.135
Well, what's this going
to be in terms of c?

00:10:06.135 --> 00:10:11.080
Well, we know that this entire
thing is a c by c square.

00:10:11.080 --> 00:10:13.920
So the area in terms
of c is just c squared.

00:10:13.920 --> 00:10:17.980
So a squared plus b squared
is equal to c squared.

00:10:17.980 --> 00:10:22.450
And we have, once again,
proven the Pythagorean theorem.