Binomial Theorem (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.070
We'll now learn an application
of combinations that you

00:00:04.070 --> 00:00:06.570
probably won't find
intuitive at first.

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But the more you think about
it, it will make a lot of sense

00:00:08.000 --> 00:00:10.820
and hopefully it'll make you
appreciate, once again, the

00:00:10.820 --> 00:00:12.140
beauty of mathematics.

00:00:12.140 --> 00:00:16.850
And then you'll also know why--
when we say n choose k in

00:00:16.850 --> 00:00:20.290
combinations --why that's also
called a binomial coefficient.

00:00:20.290 --> 00:00:23.480
Because we are going to
cover the binomial theorem.

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So before I give you the
binomial theorem, let's

00:00:26.430 --> 00:00:28.970
understand the motivation for
why it's even interesting.

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So let me erase that.

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Invert colors.

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So if we just had to
multiply-- I don't know.

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Well let's just take different
powers of a binomial.

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A binomial is just a polynomial
with two terms, right?

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So a plus b-- Well a
plus b to the 0, that's

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equal to 1, right?

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Anything to the 0
is equal to 1.

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a plus b to the 1, well
that equals a plus b.

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a plus b squared, that equals--
and if you don't have a lot of

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practice doing this you
might be tempted to say a

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squared plus b squared.

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But you should correct yourself
quickly and slap yourself on

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the wrist or the brain or
someplace if you did that.

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Because that's a plus
b times a plus b.

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And then we could use the
distributive property, or,

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if you learned it this way
in Algebra I, you could

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use the FOIL property.

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That equals a times
a plus b, right?

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Plus b times a plus b.

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Which is equal to a squared
plus ab plus ba plus b squared.

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Which is equal to a squared
plus 2ab plus b squared.

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That should be a bit
of review for you.

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And now it gets a little
bit more interesting.

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What is-- Let me circle that
just so we remember it.

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That's a plus b squared.

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What's a plus b to the third?

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And now this is starting
to get complicated.

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This is equal to a plus b times
a plus b times a plus b.

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Or another way to view it,
it's a plus b squared

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times a plus b, right?

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This is a power of three.

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So this was a plus b squared.

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So if we multiply it by
a plus b, we'll get a

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plus b to the third.

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So let's do that.

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Let's multiply this
times a plus b.

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So first let's multiply
everything times b.

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So that's b-- let me do
this in another color.

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a squared b, right?

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That's a squared times b.

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Now let's do 2ab times b.

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So plus 2ab squared, right?

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2ab times b.

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And then plus b cubed.

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And then we have a times a.

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Well that's a cubed, right?

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None of these match that, so
I'll put it in another column.

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a times 2ab.

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Well that's 2a squared b.

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I'll put that out
here: 2a squared b.

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And then a times b squared.

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Well that's plus ab
squared, right?

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And now we'll just add
up all of the terms.

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All we do is the distributive
property again, right?

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We multiplied a times all of
these terms and then added that

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to b times all of these terms.

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If we add it all up-- I'll
try to do it in order

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of-- Let's see.

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Let's put the a cubed first.

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And then-- Well actually we
already had this thing.

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This 2a squared b, I could
have written it here.

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2a squared b, because I had
an a squared b here so.

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I just rewrote the
2a squared b here.

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So we have a cubed plus 2a
squared b plus a squared b.

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That's 3a squared b.

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And then 2ab squared
plus ab squared.

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That's 2ab squared.

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And then plus b cubed.

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As you can see, that involved
a lot just to take something

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to the third power.

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So we could-- If we had the
time, we could figure out what

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a plus b to the fourth power is
or what a plus b to

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the tenth power is.

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But as you could imagine,
this would take you all day.

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So wouldn't it be neat if there
were an easy way to calculate

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what a binomial is to
an arbitrary power?

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And that's where the binomial
theorem comes into play.

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And in this video I'm going
to show you what the

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binomial theorem is.

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I will show you
how to apply it.

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I will show you a trick or
a technique that will make

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you seem like a genius.

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And then in the next video
I'll hopefully give you

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some intuition for why the
binomial theorem actually

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involves combinations.

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Why it involves actually the
binomial coefficient at all.

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So what is the
binomial theorem?

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Let me erase all of this.

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And you can confirm that the
binomial theorem works for the

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ones that we've worked out,
up to a plus b to the third.

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You could work out a plus
b to the fourth if you

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like to punish yourself.

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Let's see.

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Clear image.

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Invert color.

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So the binomial theorem tells
us that a plus b to the nth

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power is equal to-- And I know
it's going to look complicated

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at first, but we'll do a couple
of examples and you'll see

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it's not that intimidating.

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It equals the sum from k
equals 0 to n, right?

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This n is the same
thing as that n.

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Of-- each term is n
choose k, right?

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We're going to keep
incrementing k up from 0 to

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n-- of x to the n minus
k times y to the k.

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I know that looks complicated
but if we do a couple of

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concrete examples, I think it
should make a reasonable

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amount of sense.

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So given-- Oh sorry.

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This is-- This isn't-- I
was copying this down.

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This should be a to the n
minus k and this should be b.

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What I had written down
before, that would be

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x plus y to the n.

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If we have a plus b to the n, n
choose k each term. a to the

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n minus k times b to the k.

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So let's apply this, a couple
of concrete examples.

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We could even switch around the
variable names if we want just

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to show you that they don't
have to be a's and b's.

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They can be anything.

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So what is a plus b-- let's do
one that we otherwise would

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have found fairly difficult-- a
plus b to the fourth power.

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Well that, the binomial theorem
tells us that, let's see, the

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first term is going to be--
Well what's n, first of

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all? n is 4 in this case.

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It equals-- Let me fill in
all the numbers actually.

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From k equals 0 to 4
of 4 choose k, right?

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Because k is what
we're incrementing.

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a to the 4 minus k.

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b to the k, right?

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I just substituted the
n into the binomial

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theorem definition.

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And what does that equal?

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Well the first term
is k equals 0.

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So that's 4 choose 0.

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So out of 4 things, I'm
going to choose 0.

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And I'll show you in the
next video why that works.

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Of a to the 4 minus k.

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Well the first term k is 0.

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So it's a to the fourth,
b to the 0, right?

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So the b, that's just 1,
so we can just ignore it.

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So what's the next term?

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Well, it's going
to be 4 choose 1.

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And now k is 1.

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So 4 minus 1 is 3.

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a to the third.

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And k is 1 now.

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We're in the-- This
is the zeroth term.

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This is the first term.

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So b to the first plus-- So as
you can see, each term we go,

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the a term, the first term,
whichever it is, it decrements.

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It starts up at the power
n, or in the fourth power.

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And then each term,
it goes down by 1.

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And then the second term,
the b term, it starts

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at the zeroth power.

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So it starts at 1.

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That's why you don't
see it there.

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And then each term
it increments up.

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So then the next one-- So I
think you see the pattern.

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It's going to be 4 choose 2, a
squared b squared plus 4 choose

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3 ab to the third
plus 4 choose 4.

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Now it'll have a to the
zero, so that's just

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1, b to the fourth.

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So we're done if we just figure
out what these binomial

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coefficients are.

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And that's where they
come from, from the

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binomial theorem.

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But we remember how to
calculate that, right?

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In general-- and hopefully you
have the intuition on this.

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You shouldn't just memorize it.

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n choose k from our
combinatorics is equal to n

00:09:01.970 --> 00:09:07.710
factorial over k factorial
divided by n minus k factorial.

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So in this case,
what's 4 choose 0?

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That equals-- I know it seems
very time consuming right now.

00:09:19.240 --> 00:09:20.570
Although it's less time
consuming than actually

00:09:20.570 --> 00:09:21.360
multiplying it out.

00:09:21.360 --> 00:09:24.210
But I'll show you trick in a
second that will amaze you.

00:09:24.210 --> 00:09:30.480
So this is equal to 4 factorial
over 0 factorial times 4

00:09:30.480 --> 00:09:37.400
factorial, right-- 4 minus 0 is
4 --a to the fourth plus 4

00:09:37.400 --> 00:09:45.680
factorial over 1 factorial
times 3 factorial, right?

00:09:45.680 --> 00:09:48.600
4 minus 1 is 3 factorial.

00:09:48.600 --> 00:09:53.870
a to the third b plus-- I know
this is getting a little

00:09:53.870 --> 00:09:56.180
tedious, but I think it's good
to completely work through

00:09:56.180 --> 00:09:58.770
one entire problem
--plus 4 choose 2.

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That's 4 factorial over
2 factorial times 2

00:10:02.680 --> 00:10:03.410
factorial, right?

00:10:03.410 --> 00:10:05.560
4 minus 2 is 2.

00:10:05.560 --> 00:10:10.570
a squared b squared
plus 4 choose 3.

00:10:10.570 --> 00:10:15.690
That's 4 factorial
over 3 factorial.

00:10:15.690 --> 00:10:17.090
4 minus 3 is 1 factorial.

00:10:21.890 --> 00:10:24.120
ab cubed.

00:10:24.120 --> 00:10:25.200
And then 4 choose 4.

00:10:25.200 --> 00:10:29.470
That's plus 4 factorial
over 4 factorial times 0

00:10:29.470 --> 00:10:31.600
factorial b to the fourth.

00:10:31.600 --> 00:10:34.730
And notice: this coefficient is
the same as that coefficient.

00:10:34.730 --> 00:10:37.370
This coeffieicnt is the same
as this coefficient and then

00:10:37.370 --> 00:10:38.600
this one's in the middle.

00:10:38.600 --> 00:10:39.890
So let's evaluate them.

00:10:39.890 --> 00:10:42.240
And I'll switch colors.

00:10:42.240 --> 00:10:44.120
So 0 factorial, in case
you don't know it, it's

00:10:44.120 --> 00:10:45.410
actually defined to be 1.

00:10:45.410 --> 00:10:47.480
Which is a little bit
non-intuitive because 1

00:10:47.480 --> 00:10:49.240
factorial is also 1.

00:10:49.240 --> 00:10:50.430
But that's just something
you should know.

00:10:50.430 --> 00:10:53.830
So 4 factorial divided by 0
factorial times 4 factorial.

00:10:53.830 --> 00:10:55.050
This is actually equal to 1.

00:10:55.050 --> 00:10:59.940
So the first term is just a to
the fourth plus 4 factorial--

00:10:59.940 --> 00:11:04.430
it's 4 times 3 times 2 times 1
--divided by 3 times 2 times 1.

00:11:04.430 --> 00:11:06.380
So that equals 4.

00:11:06.380 --> 00:11:10.820
4a cubed b plus 4 factorial.

00:11:10.820 --> 00:11:14.580
That's 4 times 3
times 2 times 1.

00:11:14.580 --> 00:11:19.340
So that's 24, right?

00:11:19.340 --> 00:11:20.610
Over-- what's 2 factorial?

00:11:20.610 --> 00:11:21.260
That's just 2.

00:11:21.260 --> 00:11:23.420
So 2 times 2 is 4.

00:11:23.420 --> 00:11:25.170
So 24 divided by 4 is 6.

00:11:25.170 --> 00:11:32.600
So 6a squared b squared plus
well 4-- This term is the

00:11:32.600 --> 00:11:33.820
same as this term, right?

00:11:33.820 --> 00:11:35.660
With just the 1 factorial
and the 3 factorial

00:11:35.660 --> 00:11:36.430
got switched around.

00:11:36.430 --> 00:11:39.340
And you might want to think
about that for a few

00:11:39.340 --> 00:11:40.530
seconds as to why that is.

00:11:40.530 --> 00:11:42.250
It should make a
little sense to you.

00:11:42.250 --> 00:11:48.030
But that is-- So it's
going to be 4ab cubed.

00:11:48.030 --> 00:11:49.030
And it makes sense, right?

00:11:49.030 --> 00:11:51.610
Because this could have
just been b plus a.

00:11:51.610 --> 00:11:53.960
A plus b and b plus a are the
same thing so it makes sense

00:11:53.960 --> 00:11:56.280
that there's a symmetry, right?

00:11:56.280 --> 00:12:00.240
That we have 4ab cubed and
we also have 4a cubed b.

00:12:00.240 --> 00:12:02.730
Ignore me if you find
that confusing.

00:12:02.730 --> 00:12:05.220
If you find it enlightening,
all the better.

00:12:05.220 --> 00:12:06.650
And then the last term.

00:12:06.650 --> 00:12:07.550
4 factorial.

00:12:07.550 --> 00:12:09.460
This term is the same
thing as this term.

00:12:09.460 --> 00:12:10.760
And we've already figured
out that equals 1.

00:12:10.760 --> 00:12:12.540
So plus b to the fourth.

00:12:12.540 --> 00:12:13.410
So I had a little symmetry.

00:12:13.410 --> 00:12:16.680
The coefficients
are 1, 4, 6, 4, 1.

00:12:16.680 --> 00:12:19.670
And I'll show you in a future
video that these are actually

00:12:19.670 --> 00:12:24.560
the terms of a Pascal Triangle,
which is another avenue

00:12:24.560 --> 00:12:26.140
to go in mathematics.

00:12:26.140 --> 00:12:28.180
But anyway, this was
an application of the

00:12:28.180 --> 00:12:29.100
binomial theorem.

00:12:29.100 --> 00:12:31.040
And I realize I've taken
12 minutes so far.

00:12:31.040 --> 00:12:34.990
So I will do more examples
in the next video.

00:12:34.990 --> 00:12:36.492
See you soon.


Binomial Theorem (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:08.300
In the last video we saw that
if you wanted to take a plus b

00:00:08.300 --> 00:00:12.000
to the nth power, and if n is
larger than, really, 2-- but

00:00:12.000 --> 00:00:15.590
really, especially larger than
3-- it is very tedious to

00:00:15.590 --> 00:00:17.930
multiply it out, essentially
using the distributive

00:00:17.930 --> 00:00:21.480
property, or doing polynomial
multiplication, or FOIL, or

00:00:21.480 --> 00:00:22.130
however you learned it.

00:00:22.130 --> 00:00:24.410
It is extremely,
extremely tedious.

00:00:24.410 --> 00:00:26.920
And then we learned that the
binomial theorem, which said

00:00:26.920 --> 00:00:34.170
this, that that is equal to
the sum from k is equal to 0,

00:00:34.170 --> 00:00:39.470
to n of n choose k-- right?

00:00:39.470 --> 00:00:43.390
Where that was what we learned
in combinatorics as the

00:00:43.390 --> 00:00:44.350
binomial coefficient.

00:00:44.350 --> 00:00:46.130
And that's why it's called a
binomial coefficient, because

00:00:46.130 --> 00:00:48.970
it's actually the coefficient
of the binomial theorem.-- Of

00:00:48.970 --> 00:00:53.570
x to the n minus k-- oh,
sorry, I keep writing x.

00:00:53.570 --> 00:00:55.210
Let me undo that.

00:00:55.210 --> 00:00:56.920
Edit, undo.

00:00:56.920 --> 00:00:59.250
Edit, undo.

00:00:59.250 --> 00:01:01.250
Oh, it's taking too long.

00:01:01.250 --> 00:01:05.810
Oh, let me just-- oh no, that's
not what I wanted to do.

00:01:05.810 --> 00:01:07.860
Let me erase it.

00:01:07.860 --> 00:01:09.086
OK.

00:01:09.086 --> 00:01:11.190
I keep writing x.

00:01:11.190 --> 00:01:12.680
It could be an x, but then
this would have to be

00:01:12.680 --> 00:01:13.650
an x here, as well.

00:01:16.570 --> 00:01:17.810
Maybe I should do that.

00:01:17.810 --> 00:01:26.780
--Of a to the n minus
k, times b to the k.

00:01:26.780 --> 00:01:29.840
So each term-- you know, the n
stays constant-- but each term,

00:01:29.840 --> 00:01:32.130
you start at k equals 0 and
you keep incrementing up.

00:01:32.130 --> 00:01:35.060
And we did an example to solve
a plus b to the fourth power

00:01:35.060 --> 00:01:37.210
in the previous video.

00:01:37.210 --> 00:01:41.840
And as you saw, that was
tedious, but less tedious than

00:01:41.840 --> 00:01:43.640
actually multiplying it out.

00:01:43.640 --> 00:01:48.610
And if you get really fast
at computing n choose k

00:01:48.610 --> 00:01:52.640
for different ns and k, it
could be reasonably fast.

00:01:52.640 --> 00:01:55.130
So what I want to do is, I'm
going to show you a slightly

00:01:55.130 --> 00:01:58.010
faster method than
what we just did.

00:01:58.010 --> 00:02:02.060
Kind of a faster way to compute
the binomial coefficients.

00:02:02.060 --> 00:02:04.480
And then after that, I'm going
to show you a super fast way

00:02:04.480 --> 00:02:07.290
that, short of memorizing the
coefficients-- which I actually

00:02:07.290 --> 00:02:11.520
know some people who've done
that-- is a pretty amazing way

00:02:11.520 --> 00:02:14.590
to essentially multiply
out any binomial.

00:02:14.590 --> 00:02:17.410
So what's my pseudo
fast way of doing it?

00:02:17.410 --> 00:02:21.820
Well, I hinted in the last
presentation that those

00:02:21.820 --> 00:02:24.640
coefficients were actually
terms of a Pascal triangle.

00:02:24.640 --> 00:02:26.720
So what's a Pascal triangle?

00:02:26.720 --> 00:02:30.430
So if we start off with a 1 and
then you just go-- actually,

00:02:30.430 --> 00:02:33.210
let me do it-- well, yeah,
let me do it right here.

00:02:33.210 --> 00:02:37.190
And then, actually let
me start with two 1s.

00:02:37.190 --> 00:02:41.340
And what you do is, you take
the sum of both of these.

00:02:41.340 --> 00:02:43.100
So that's a 2.

00:02:43.100 --> 00:02:47.190
And then you bring down
a 1, to the left and

00:02:47.190 --> 00:02:48.450
the right hand side.

00:02:48.450 --> 00:02:50.620
And notice, these are
the coefficients of

00:02:50.620 --> 00:02:51.740
a plus b squared.

00:02:55.330 --> 00:02:58.100
And these are the
coefficients of a plus b.

00:02:58.100 --> 00:03:00.530
You could say a
plus b to the one.

00:03:00.530 --> 00:03:02.480
1a plus 1b.

00:03:02.480 --> 00:03:06.980
This is a squared-- so you
could rewrite, a plus b

00:03:06.980 --> 00:03:14.070
squared. is 1a squared
plus 2ab plus 1b squared.

00:03:14.070 --> 00:03:17.400
So these are the coefficients
of a plus b squared.

00:03:17.400 --> 00:03:20.110
Let me arbitrarily
switch colors.

00:03:20.110 --> 00:03:24.090
And so, 1 plus 2 is 3.

00:03:24.090 --> 00:03:26.290
2 plus 1 is 3.

00:03:26.290 --> 00:03:28.010
Bring down the 1.

00:03:28.010 --> 00:03:29.610
Bring down the 1.

00:03:29.610 --> 00:03:31.560
And now we have the
coefficients for a

00:03:31.560 --> 00:03:32.720
plus b to the third.

00:03:32.720 --> 00:03:35.570
Which we computed in
the-- that was the very

00:03:35.570 --> 00:03:36.150
first thing we did.

00:03:36.150 --> 00:03:37.250
We actually multiplied it out.

00:03:37.250 --> 00:03:39.520
And we just know the pattern.

00:03:39.520 --> 00:03:41.150
The first coefficient is 1.

00:03:41.150 --> 00:03:46.780
So it's 1a to the third, b to
the zero-- so we don't have

00:03:46.780 --> 00:03:50.200
to write the b-- plus 3.

00:03:50.200 --> 00:03:52.680
We just decrement
this exponent 1.

00:03:52.680 --> 00:04:03.370
3a squared b plus 3ab squared,
and then plus 1a to the zero--

00:04:03.370 --> 00:04:06.680
which is just 1-- b cubed.

00:04:06.680 --> 00:04:07.790
So that was pretty fast.

00:04:07.790 --> 00:04:10.260
And we can keep going down
the Pascal triangle.

00:04:10.260 --> 00:04:11.890
So let's do the next one.

00:04:11.890 --> 00:04:13.760
So we can bring down a 1.

00:04:13.760 --> 00:04:16.750
1 plus 3 is 4.

00:04:16.750 --> 00:04:18.220
3 plus 3 is 6.

00:04:18.220 --> 00:04:19.080
And this is neat.

00:04:19.080 --> 00:04:22.400
I mean, just very simply, you
can actually generate binomial

00:04:22.400 --> 00:04:24.990
coefficients without
having to compute them.

00:04:24.990 --> 00:04:27.160
Very simple, I guess you
could call it an algorithm.

00:04:27.160 --> 00:04:29.750
Or drawing.

00:04:29.750 --> 00:04:31.970
And it's symmetric, just as
you would expect it, right?

00:04:31.970 --> 00:04:34.750
Because you could
easily switch b and a.

00:04:34.750 --> 00:04:36.540
a plus b is the same thing
as b plus a, so you

00:04:36.540 --> 00:04:38.620
should essentially
get the same answer.

00:04:38.620 --> 00:04:40.920
And so, we just-- very quickly
we figured out the binomial

00:04:40.920 --> 00:04:42.770
coefficients for a
plus b to the fourth.

00:04:42.770 --> 00:04:46.990
Which was a lot faster than
we did in the last example.

00:04:46.990 --> 00:04:48.660
a plus b to the fourth.

00:04:48.660 --> 00:04:50.700
So then we-- I think
you get the point.

00:04:50.700 --> 00:04:54.660
But, so it's 1-- let me write
in a different color-- 1a

00:04:54.660 --> 00:05:00.080
to the fourth, b to the
zero, plus 4a cubed b

00:05:00.080 --> 00:05:03.400
squared-- b to the one.

00:05:03.400 --> 00:05:07.930
Plus 6a squared b squared.

00:05:07.930 --> 00:05:10.640
Which makes sense that, you
know, this is a middle number

00:05:10.640 --> 00:05:12.890
and they both-- and a and b
have the same exponent

00:05:12.890 --> 00:05:14.150
at this point.

00:05:14.150 --> 00:05:19.490
And then plus 4a-- we
decrement it-- b cubed.

00:05:19.490 --> 00:05:20.640
Plus b to the fourth.

00:05:20.640 --> 00:05:22.946
1b to the fourth, right? a
to the zero, so that's what

00:05:22.946 --> 00:05:23.780
we didn't write there.

00:05:23.780 --> 00:05:26.620
So 1b to the fourth.

00:05:26.620 --> 00:05:28.940
And that was very fast compared
to what we had to do at

00:05:28.940 --> 00:05:30.420
the end of the last video.

00:05:30.420 --> 00:05:32.460
We could just keep going.

00:05:32.460 --> 00:05:33.990
You know, for 5.

00:05:33.990 --> 00:05:37.980
So 1 plus 4 is five.

00:05:37.980 --> 00:05:40.250
4 plus 6 is 10.

00:05:40.250 --> 00:05:42.810
6 plus 4 is 10.

00:05:42.810 --> 00:05:44.910
4 plus 1 is 5.

00:05:44.910 --> 00:05:45.620
Bring down the 1.

00:05:45.620 --> 00:05:49.240
So these are the coefficients
for the expansion of a

00:05:49.240 --> 00:05:51.910
plus b to the fifth power.

00:05:51.910 --> 00:05:54.360
And so this is a reasonably
fast way of doing it, although

00:05:54.360 --> 00:05:59.040
it can get-- one, it will
take a lot of space.

00:05:59.040 --> 00:06:05.270
And it can work reasonably
well, you know, for up to a

00:06:05.270 --> 00:06:08.030
power of eight or nine or ten.

00:06:08.030 --> 00:06:10.490
Even then it starts to get
pretty big and cumbersome.

00:06:10.490 --> 00:06:13.080
But you know, for powers
up to seven or eight or

00:06:13.080 --> 00:06:14.450
nine, you could do this.

00:06:14.450 --> 00:06:17.400
You could draw it out really
fast and do this, and it's

00:06:17.400 --> 00:06:19.950
probably faster than actually
computing each of the

00:06:19.950 --> 00:06:20.830
binomial coefficients.

00:06:20.830 --> 00:06:23.590
Although you might be pretty
fast at computing n choose k,

00:06:23.590 --> 00:06:25.580
in which you don't
have to do this.

00:06:25.580 --> 00:06:29.230
So with that out of the way,
let me show you an even faster

00:06:29.230 --> 00:06:31.200
way of doing it, short
of memorizing it.

00:06:31.200 --> 00:06:33.810
And this will allow you to
really calculate a plus b

00:06:33.810 --> 00:06:37.200
to the nth-- you know, to
the twentieth power--

00:06:37.200 --> 00:06:38.030
almost in your head.

00:06:38.030 --> 00:06:41.610
Depending on how good you are
at arithmetic in your head.

00:06:41.610 --> 00:06:42.850
So here is the trick.

00:06:42.850 --> 00:06:45.800
And I encourage you to
experiment for why it

00:06:45.800 --> 00:06:47.720
works, but it does work.

00:06:47.720 --> 00:06:49.580
And I mean it's
not even a trick.

00:06:49.580 --> 00:06:54.340
It's just-- and this Pascal's
triangle isn't even a trick.

00:06:54.340 --> 00:06:56.960
Pascal's triangle is just an
alternative way to generate

00:06:56.960 --> 00:07:00.480
binomial coefficients, and what
I'm about to show you is just

00:07:00.480 --> 00:07:03.380
another way of essentially
generating the binomial

00:07:03.380 --> 00:07:03.960
coefficients.

00:07:03.960 --> 00:07:06.301
Although it's probably a
faster way to compute them.

00:07:10.450 --> 00:07:14.690
And it's a good project for you
to think about why this works.

00:07:14.690 --> 00:07:17.010
So I'm just going to start
with a very concrete example.

00:07:17.010 --> 00:07:19.020
Instead of a plus b, let
me just do x plus y.

00:07:19.020 --> 00:07:23.320
Just because you might
see the binomial theorem

00:07:23.320 --> 00:07:24.140
written that way.

00:07:24.140 --> 00:07:26.160
So let's say x plus y
to the tenth power.

00:07:26.160 --> 00:07:30.100
This would take me all
day if I was to actually

00:07:30.100 --> 00:07:32.150
multiply it out.

00:07:32.150 --> 00:07:36.570
It would take me probably 20
to 30 minutes, without

00:07:36.570 --> 00:07:38.720
making careless mistakes, to
actually figure out all the

00:07:38.720 --> 00:07:39.680
binomial coefficients.

00:07:39.680 --> 00:07:41.670
Maybe not that long, but
it would take me a while.

00:07:41.670 --> 00:07:44.130
And to draw Pascal's triangle
would fill up a whole page,

00:07:44.130 --> 00:07:45.570
and I'd still probably
make a careless mistake.

00:07:45.570 --> 00:07:46.990
So how can I do this?

00:07:46.990 --> 00:07:50.700
So what you do is-- so
one thing you know.

00:07:50.700 --> 00:07:53.260
This is going to have
11 terms, right?

00:07:53.260 --> 00:07:55.160
Because you're going to
start with x to the

00:07:55.160 --> 00:07:56.435
tenth, y to the zero.

00:07:56.435 --> 00:07:59.230
And you're going to go all
the way to y to the tenth.

00:07:59.230 --> 00:08:02.602
So if you start at 0 and you
go to 10, that's 11 terms.

00:08:02.602 --> 00:08:03.790
So it has 11 terms.

00:08:03.790 --> 00:08:06.340
What I want you to do
is just write down the

00:08:06.340 --> 00:08:07.700
first, just the numbers.

00:08:07.700 --> 00:08:09.210
You know, you can almost
count the terms.

00:08:09.210 --> 00:08:10.820
You don't have to go all the
way to 11, and I'll show you.

00:08:10.820 --> 00:08:12.830
But actually, let's write
all the way to 11.

00:08:12.830 --> 00:08:26.010
So 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11.

00:08:26.010 --> 00:08:26.780
Just squeezed it in.

00:08:26.780 --> 00:08:28.250
And you'll see, you don't
actually have to go

00:08:28.250 --> 00:08:28.770
all the way to 11.

00:08:28.770 --> 00:08:30.680
You could probably
just stop at 6.

00:08:30.680 --> 00:08:31.450
So here's the trick.

00:08:31.450 --> 00:08:34.430
We know that the first
term is going to be x

00:08:34.430 --> 00:08:37.350
to the tenth, right?

00:08:37.350 --> 00:08:39.100
We know that x to the tenth.

00:08:39.100 --> 00:08:41.310
Actually, we know that it's
going to be x to the tenth.

00:08:41.310 --> 00:08:43.320
The second term is going
to be x to the ninth.

00:08:43.320 --> 00:08:45.680
Then it's going to
be x to the eighth.

00:08:45.680 --> 00:08:47.510
It's going to be x
to the seventh.

00:08:47.510 --> 00:08:49.630
A little tedious.
x to the sixth.

00:08:49.630 --> 00:08:51.080
x to the fifth.

00:08:51.080 --> 00:08:52.735
x to the fourth.

00:08:52.735 --> 00:08:54.400
x to the third.

00:08:54.400 --> 00:08:55.790
x squared.

00:08:55.790 --> 00:08:56.750
x.

00:08:56.750 --> 00:09:00.060
And then there's going to be
x to the zero, or just one.

00:09:00.060 --> 00:09:03.970
Let me just do the y's.

00:09:03.970 --> 00:09:07.410
So this was x to the tenth.

00:09:07.410 --> 00:09:10.400
That's not bright
enough, this color.

00:09:10.400 --> 00:09:12.080
So this is y to the zero.

00:09:12.080 --> 00:09:12.910
So we don't have to
write it there.

00:09:12.910 --> 00:09:15.230
But then we have a y.

00:09:15.230 --> 00:09:16.770
y to the first.

00:09:16.770 --> 00:09:19.650
y squared.

00:09:19.650 --> 00:09:22.370
y to the third.

00:09:22.370 --> 00:09:24.120
y to the fourth.

00:09:24.120 --> 00:09:25.320
y to the fifth.

00:09:25.320 --> 00:09:27.470
Which makes sense, this
is the middle term.

00:09:27.470 --> 00:09:29.750
y to the sixth.

00:09:29.750 --> 00:09:31.830
y to the seventh.

00:09:31.830 --> 00:09:33.750
y to the eighth.

00:09:33.750 --> 00:09:34.860
y to the ninth.

00:09:34.860 --> 00:09:36.300
I don't want you to get
confused, each of these

00:09:36.300 --> 00:09:37.120
is a separate term.

00:09:37.120 --> 00:09:38.890
I don't want you to think
I'm multiplying them all.

00:09:43.420 --> 00:09:46.010
And then, we just have to
figure out the coefficients

00:09:46.010 --> 00:09:46.820
on each of these terms.

00:09:46.820 --> 00:09:48.500
Those are divider lines
I attempted to draw.

00:09:48.500 --> 00:09:50.350
I wasn't trying to
confuse you more.

00:09:50.350 --> 00:09:52.970
I just wanted to-- because they
seemed to be running together,

00:09:52.970 --> 00:09:54.770
each of the terms
that I'm writing.

00:09:54.770 --> 00:09:56.130
But I think you know
what I'm doing.

00:09:56.130 --> 00:09:57.570
So now we have to figure
out the coefficients.

00:09:57.570 --> 00:10:00.100
And then this is the neat part.

00:10:00.100 --> 00:10:02.250
So we know that the coefficient
on the first term-- let me draw

00:10:02.250 --> 00:10:05.370
a dividing line here and here--
the coefficient on the first

00:10:05.370 --> 00:10:06.710
term is always 1, right?

00:10:06.710 --> 00:10:08.570
So, the coefficient is 1.

00:10:08.570 --> 00:10:12.110
So the coefficient on the
second term is going to be the

00:10:12.110 --> 00:10:16.750
exponent on the first term
times its coefficient-- so 10

00:10:16.750 --> 00:10:19.670
times 1-- divided by
the term that it is.

00:10:19.670 --> 00:10:23.510
So it's going to be 10
times 1, divided by 1.

00:10:23.510 --> 00:10:25.900
So it's going to be 10.

00:10:25.900 --> 00:10:27.720
The third term's coefficient
is going to be the

00:10:27.720 --> 00:10:30.180
exponent on the x, right?

00:10:30.180 --> 00:10:34.980
So it's 9 times its
coefficient-- which is 10-- so

00:10:34.980 --> 00:10:40.040
it's going to be 9 times 10--
divided by the term it is.

00:10:40.040 --> 00:10:45.510
So it's going to be 9 times its
coefficient, 10, divided by 2.

00:10:45.510 --> 00:10:46.480
So what's 9 times 10?

00:10:46.480 --> 00:10:51.440
That's forty-- it's 90
divided by 2 which is 45.

00:10:51.440 --> 00:10:52.640
And you keep going.

00:10:52.640 --> 00:10:56.700
The fourth term is going to be
the third term's exponent-- so

00:10:56.700 --> 00:11:00.140
it's going to be 8 times-- let
me write this down in a

00:11:00.140 --> 00:11:06.840
different color-- it's going to
be 8 times its coefficient,

00:11:06.840 --> 00:11:11.140
times 45 divided by
which term it is.

00:11:11.140 --> 00:11:12.050
So it's the third term.

00:11:12.050 --> 00:11:13.610
Divided by 3.

00:11:13.610 --> 00:11:16.740
Well, that's just 8 times 15.

00:11:16.740 --> 00:11:19.060
And we'll see,
that's 80 plus 40.

00:11:19.060 --> 00:11:20.920
So that's equal to 120.

00:11:20.920 --> 00:11:23.730
So that is the fourth term.

00:11:23.730 --> 00:11:25.745
And so then let me just
draw these dividers.

00:11:25.745 --> 00:11:27.630
I know it's getting a
little complicated.

00:11:27.630 --> 00:11:30.310
And I'm writing it all out like
this, but if you practice

00:11:30.310 --> 00:11:32.510
this enough, you can actually
just write it straight out.

00:11:32.510 --> 00:11:34.060
And so the fifth term.

00:11:34.060 --> 00:11:35.230
What is the fifth term?

00:11:35.230 --> 00:11:36.910
Well, you take the
exponent on the x.

00:11:36.910 --> 00:11:42.670
So 7 times the fourth
term's coefficient--

00:11:42.670 --> 00:11:46.960
times 120-- divided by 4.

00:11:46.960 --> 00:11:47.220
Right?

00:11:47.220 --> 00:11:50.040
Divided by the
previous term, by 4.

00:11:50.040 --> 00:11:55.950
Well, that's just 7 times
30, which equals 210.

00:11:55.950 --> 00:11:57.900
That's the fifth coefficient.

00:11:57.900 --> 00:12:00.010
What's the sixth coefficient?

00:12:00.010 --> 00:12:06.280
Well, it's 6 times-- you
know the exponent on

00:12:06.280 --> 00:12:10.160
the x-- times 210.

00:12:10.160 --> 00:12:13.780
Times its coefficient-- times
the fifth term's coefficient--

00:12:13.780 --> 00:12:16.830
divided by 5, for
the fifth term.

00:12:16.830 --> 00:12:18.780
Well, 5 goes into
210 how many times?

00:12:18.780 --> 00:12:21.100
42 times, right?

00:12:21.100 --> 00:12:25.020
So it's 6 times 42--
that's 240 plus 12.

00:12:25.020 --> 00:12:30.250
That's 252.

00:12:30.250 --> 00:12:33.590
And then once you're at the
middle point-- the sixth term

00:12:33.590 --> 00:12:36.220
is the middle term-- you'll
see that, you know, you start

00:12:36.220 --> 00:12:37.130
going back the other way.

00:12:37.130 --> 00:12:39.360
And we learned one from the
Pascal's triangle, or even the

00:12:39.360 --> 00:12:42.240
definition of the binomial
theorem, that the

00:12:42.240 --> 00:12:43.450
coefficients are symmetric.

00:12:43.450 --> 00:12:45.830
So we know that the next one is
going to be the same-- this was

00:12:45.830 --> 00:12:48.040
the middle one, right?-- so we
know the next one is

00:12:48.040 --> 00:12:51.340
going to be 210.

00:12:51.340 --> 00:12:53.620
And you could calculate it
using the same system.

00:12:53.620 --> 00:12:55.020
This is just a quick
way of doing it.

00:12:55.020 --> 00:12:56.680
This one's going to be 120.

00:12:56.680 --> 00:12:58.800
This one's going to be 45.

00:12:58.800 --> 00:13:04.880
And this one is going to be
the tenth coefficient--

00:13:04.880 --> 00:13:05.860
this is going to be 10.

00:13:05.860 --> 00:13:07.640
And then of course the last
coefficient is just 1.

00:13:07.640 --> 00:13:10.210
1y to the tenth power.

00:13:10.210 --> 00:13:15.510
So if I were to write this out,
the answer is-- and if you

00:13:15.510 --> 00:13:18.870
practice this, you'll find that
you can do quite fast-- it's x

00:13:18.870 --> 00:13:35.280
to the tenth, plus 10x to the
ninth y, plus 45x to the eighth

00:13:35.280 --> 00:13:43.500
y squared, plus 120x to the
seventh y to the third, plus

00:13:43.500 --> 00:13:49.910
210x to the sixth y to the
fourth, plus 252-- already at

00:13:49.910 --> 00:13:56.520
the middle term-- x to the
fifth y to the fifth, plus 210x

00:13:56.520 --> 00:13:59.460
to the fourth y to the sixth--
I'm running out of space.

00:13:59.460 --> 00:14:02.070
But you can you can hopefully
extrapolate what I'm doing,

00:14:02.070 --> 00:14:03.100
and it makes sense to you.

00:14:03.100 --> 00:14:05.620
And hopefully you have an
appreciation that if you

00:14:05.620 --> 00:14:08.346
actually had to multiply out x
plus y to the tenth, it would

00:14:08.346 --> 00:14:09.960
have taken you all day.

00:14:09.960 --> 00:14:13.213
Maybe I'll do one more video
with a smaller example to show

00:14:13.213 --> 00:14:15.620
you that it's a little less
complicated when you do, say,

00:14:15.620 --> 00:14:17.260
x plus y to the sixth power.

00:14:17.260 --> 00:14:18.762
See you soon.


Conditional probability and combinations

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:01.440
Welcome back.

00:00:01.440 --> 00:00:04.430
Now let's do a problem that
involves almost everything

00:00:04.430 --> 00:00:07.460
we've learned so far about
probability and combinations

00:00:07.460 --> 00:00:08.880
and conditional probability.

00:00:08.880 --> 00:00:10.485
So let's say I
have a bag again.

00:00:10.485 --> 00:00:17.700
And in that bag, I have
5 fair coins, and I

00:00:17.700 --> 00:00:19.060
have 10 unfair coins.

00:00:21.960 --> 00:00:24.580
And a fair coin, of course, is
a 50:50 chance of getting heads

00:00:24.580 --> 00:00:28.930
or tails, and the unfair coin--
let's say that there is an 80%

00:00:28.930 --> 00:00:33.240
chance of getting a heads for
any one of those coins,

00:00:33.240 --> 00:00:36.110
and that there is a 20%
chance of getting tails.

00:00:36.110 --> 00:00:36.350
Right?

00:00:36.350 --> 00:00:39.020
Because it's going to
either be heads or tails.

00:00:39.020 --> 00:00:43.160
So my question is, what happens
is I put my hand in the bag,

00:00:43.160 --> 00:00:47.430
and my eyes are closed,
and I picked out a coin.

00:00:47.430 --> 00:00:49.055
And then I flip it six times.

00:00:55.090 --> 00:00:59.380
Let's say I got four
out of six heads.

00:00:59.380 --> 00:01:01.620
That's the result I got.

00:01:01.620 --> 00:01:05.530
What I want to know is, what is
the probability that I picked

00:01:05.530 --> 00:01:12.040
out a fair coin, given that I
got four out of six heads?

00:01:14.550 --> 00:01:17.430
So before moving on, let's do a
little bit of review of Bayes'

00:01:17.430 --> 00:01:19.490
Theorem, and I think that'll
give us a good framework for

00:01:19.490 --> 00:01:21.090
the rest of this problem.

00:01:21.090 --> 00:01:24.760
So Bayes' Theorem-- and let me
do it in this corner up here.

00:01:24.760 --> 00:01:28.210
Bayes' Theorem tells us
the probability of both

00:01:28.210 --> 00:01:30.610
a and b happening.

00:01:30.610 --> 00:01:33.250
That upside down u is just an
intersection in set theory, but

00:01:33.250 --> 00:01:35.930
it's essentially saying, you
know, it's a set of events in

00:01:35.930 --> 00:01:38.340
which both a and b occur.

00:01:38.340 --> 00:01:43.810
That's equal to the probability
of a occurring given b, times

00:01:43.810 --> 00:01:49.660
the probability of b, which is
also equal to the probability

00:01:49.660 --> 00:01:55.655
of b occurring given a,
times the probability of a.

00:01:55.655 --> 00:01:58.450
And I think this should make
some intuition for you.

00:01:58.450 --> 00:02:01.570
If it doesn't, it might be
a good idea to watch the

00:02:01.570 --> 00:02:03.450
conditional probability videos.

00:02:03.450 --> 00:02:06.970
But what we can do is we can
rearrange this equation right

00:02:06.970 --> 00:02:10.115
here to get-- if we just divide
both sides by the probability

00:02:10.115 --> 00:02:13.280
of b, we get the probability--
and I'll do this in a vibrant

00:02:13.280 --> 00:02:19.910
color-- the probability of a
given b is equal to the

00:02:19.910 --> 00:02:27.770
probability of b given a, times
the probability of a divided

00:02:27.770 --> 00:02:31.530
by the probability of b.

00:02:31.530 --> 00:02:33.530
I just took this equation,
divided both sides by the

00:02:33.530 --> 00:02:35.560
probability of b,
and I got this.

00:02:35.560 --> 00:02:38.250
So what is a and b in the
problem we're trying

00:02:38.250 --> 00:02:38.610
to figure out?

00:02:38.610 --> 00:02:40.470
We're want to try to figure out
the probability that I picked

00:02:40.470 --> 00:02:43.810
out a fair coin, given that I
got four out of six heads.

00:02:43.810 --> 00:02:47.250
So in this situation, a is
that I got a fair coin.

00:02:47.250 --> 00:02:49.850
a is equal to picked fair coin.

00:02:55.090 --> 00:02:59.450
And then b is equal to
four out of six heads.

00:03:03.240 --> 00:03:06.440
So in order to figure out the
probability that I picked a

00:03:06.440 --> 00:03:08.360
fair coin, given that I got
four out of six heads, I have

00:03:08.360 --> 00:03:10.620
to know the probability of
getting four out of six heads

00:03:10.620 --> 00:03:14.120
given that I picked the fair
coin, times the probability of

00:03:14.120 --> 00:03:16.870
picking out a fair coin,
divided by the probability of

00:03:16.870 --> 00:03:19.250
getting four out of six
heads, in general.

00:03:19.250 --> 00:03:21.990
So this is probably the
hardest part to figure out.

00:03:21.990 --> 00:03:24.710
And we will-- along the way--
we will actually probably

00:03:24.710 --> 00:03:26.830
figure out the top 2 terms.

00:03:26.830 --> 00:03:29.680
So what's the probability of b,
or the probability of getting

00:03:29.680 --> 00:03:31.800
four out of six heads?

00:03:31.800 --> 00:03:33.290
Let's see what happens.

00:03:33.290 --> 00:03:38.110
Right when I put my hand into
the bag and I pick out a coin,

00:03:38.110 --> 00:03:42.470
there is a five in fifteen
chance-- right, there are 15

00:03:42.470 --> 00:03:44.630
total coins-- that I
pick a fair coin.

00:03:44.630 --> 00:03:47.550
So five in fifteen-- that's
the same thing as 1/3--

00:03:47.550 --> 00:03:49.440
that I pick a fair coin.

00:03:49.440 --> 00:03:56.860
And then there's a 2/3 chance
that I pick an unfair coin.

00:03:56.860 --> 00:04:02.730
Now if I pick a fair coin,
given that I have a fair coin,

00:04:02.730 --> 00:04:05.710
what is the probability
given the fair coin?

00:04:05.710 --> 00:04:08.520
What is the probability that
I get four out of six heads?

00:04:11.420 --> 00:04:13.460
Well, once again, let's
think about the previous

00:04:13.460 --> 00:04:15.080
several videos.

00:04:15.080 --> 00:04:17.690
What's the probability of
getting any one particular

00:04:17.690 --> 00:04:19.150
combination of four
out of six heads?

00:04:19.150 --> 00:04:23.950
So, for example, you know,
it could be heads, tails,

00:04:23.950 --> 00:04:27.585
heads, tails, heads, heads.

00:04:30.500 --> 00:04:33.380
It could be the first four
heads: heads, heads, heads,

00:04:33.380 --> 00:04:35.560
heads, tails, tails.

00:04:35.560 --> 00:04:35.750
Right?

00:04:35.750 --> 00:04:39.890
And there are a bunch of these,
and we once again will use the

00:04:39.890 --> 00:04:42.050
binomial coefficient, or we'll
use our knowledge of

00:04:42.050 --> 00:04:43.450
combinations to figure
out how many different

00:04:43.450 --> 00:04:44.340
combinations there are.

00:04:44.340 --> 00:04:47.040
But what's the probability of
each of these combinations?

00:04:47.040 --> 00:04:49.040
Well, what's the
probability of heads?

00:04:49.040 --> 00:04:53.030
That's .5 times .5
times .5 times .5.

00:04:53.030 --> 00:04:54.330
And the probability of
tails, if this is a fair coin,

00:04:54.330 --> 00:04:56.460
is also .5 times .5 times .5.

00:04:56.460 --> 00:04:59.660
So each of these-- there's a
1/2 chance of getting a heads,

00:04:59.660 --> 00:05:02.510
times a 1/2 chance of a tails,
times a 1/2 chance of a heads,

00:05:02.510 --> 00:05:04.600
times 1/2 chance of a tails,
et cetera, et cetera.

00:05:04.600 --> 00:05:06.990
So each of these are
essentially 1/2 times

00:05:06.990 --> 00:05:08.910
1/2, six times.

00:05:08.910 --> 00:05:11.070
So the probability of each
of the combinations is

00:05:11.070 --> 00:05:12.380
1/2 to the sixth power.

00:05:15.830 --> 00:05:19.190
And so how many combinations
are there like this?

00:05:19.190 --> 00:05:24.370
Where you get-- out of the six
flips, you're essentially

00:05:24.370 --> 00:05:26.150
choosing four heads.

00:05:26.150 --> 00:05:29.550
You're choosing-- if I'm once
again the God of probability--

00:05:29.550 --> 00:05:34.550
I'm picking four, exactly
four, of the six heads.

00:05:34.550 --> 00:05:37.630
Sorry, I'm picking four of
exactly six of the flips

00:05:37.630 --> 00:05:38.940
to end up heads, right?

00:05:38.940 --> 00:05:44.060
I'm choosing which of the flips
get selected, so to speak.

00:05:44.060 --> 00:05:50.480
So it's essentially, there are
going to be-- out of six flips,

00:05:50.480 --> 00:05:54.090
I'm choosing, as the God of
probability-- four to be heads.

00:05:54.090 --> 00:05:59.230
So that's the number of unique
combinations, where you have

00:05:59.230 --> 00:06:02.120
four out of six heads, times
the probability of each of

00:06:02.120 --> 00:06:06.670
the combinations, which is
1/2 to the sixth power.

00:06:06.670 --> 00:06:08.610
Well, what's 6 choose 4?

00:06:08.610 --> 00:06:14.830
That's 6 factorial over
4 factorial times 6

00:06:14.830 --> 00:06:15.880
minus 4 factorial.

00:06:15.880 --> 00:06:16.840
So that's 2 factorial.

00:06:19.630 --> 00:06:23.620
And that's times
1/2 to the sixth.

00:06:23.620 --> 00:06:25.530
I'm going to switch
colors again, just to

00:06:25.530 --> 00:06:27.370
stop the monotony.

00:06:27.370 --> 00:06:33.580
And that equals 6 times 5 times
4 times 3 times 2-- we don't

00:06:33.580 --> 00:06:36.020
have to write the 1 times 1,
but I'll do it anyway--

00:06:36.020 --> 00:06:37.720
over 4 factorial.

00:06:37.720 --> 00:06:40.980
4 times 3 times 2 times 1.

00:06:40.980 --> 00:06:41.930
And then 2 factorial.

00:06:41.930 --> 00:06:43.490
2 times 1.

00:06:43.490 --> 00:06:45.320
So that cancels with that.

00:06:45.320 --> 00:06:47.850
The 1 we can ignore.

00:06:47.850 --> 00:06:50.530
2, divide the numerator
and denominator by 2,

00:06:50.530 --> 00:06:51.950
and this becomes 3.

00:06:51.950 --> 00:06:53.560
So this becomes 15.

00:06:53.560 --> 00:06:58.720
So this equals 15 times
1/2 to the sixth.

00:06:58.720 --> 00:06:59.690
What's 1/2 to the sixth?

00:06:59.690 --> 00:07:02.550
It's 1/64, right?

00:07:02.550 --> 00:07:06.770
So 1/64, so it becomes 15/64.

00:07:06.770 --> 00:07:10.590
So the probability of getting
four out of six heads, given a

00:07:10.590 --> 00:07:12.190
fair coin, is 15 out of 64.

00:07:19.740 --> 00:07:23.225
And if you look at it, based on
our definition of b and a, this

00:07:23.225 --> 00:07:26.560
is the probability
of b given a.

00:07:26.560 --> 00:07:30.590
Right? b is four out of six
heads, given a fair coin.

00:07:30.590 --> 00:07:31.850
Fair enough.

00:07:31.850 --> 00:07:35.600
So let's figure out the
probability of-- because

00:07:35.600 --> 00:07:37.690
there's two ways of getting
four out of six heads.

00:07:37.690 --> 00:07:40.710
One, that we picked a
fair coin, and then

00:07:40.710 --> 00:07:43.070
times 15 out of 64.

00:07:43.070 --> 00:07:44.960
And then there's the
probability that we

00:07:44.960 --> 00:07:47.080
picked an unfair coin.

00:07:47.080 --> 00:07:49.490
So what's the probability
of the unfair coin?

00:07:49.490 --> 00:07:53.670
Of getting four out of six
heads, given the unfair coin?

00:07:53.670 --> 00:07:55.930
Well, once again, what's the
probability of each of the

00:07:55.930 --> 00:07:57.410
combinations where we
got four out of six?

00:07:57.410 --> 00:07:59.740
So in this situation,
let's do the same one.

00:07:59.740 --> 00:08:04.680
Heads, tails, heads,
tails, heads, heads.

00:08:04.680 --> 00:08:05.860
That's four out of six heads.

00:08:05.860 --> 00:08:08.930
But in this situation, it's not
a 50% chance of getting heads.

00:08:08.930 --> 00:08:09.630
It's 80%.

00:08:09.630 --> 00:08:16.090
So it would be .8 times
.2 times .8 times .2

00:08:16.090 --> 00:08:19.170
times .8 times .8.

00:08:19.170 --> 00:08:21.410
Now, essentially, we have-- you
know, this multiplication, we

00:08:21.410 --> 00:08:23.060
can rearrange it, because it
doesn't matter what order

00:08:23.060 --> 00:08:24.290
you multiply things in.

00:08:24.290 --> 00:08:32.470
So it's .8 to the fourth
power times .2 squared.

00:08:32.470 --> 00:08:33.080
And it doesn't matter.

00:08:33.080 --> 00:08:37.050
You know, any of the unique
combinations will each have

00:08:37.050 --> 00:08:37.910
the same probability.

00:08:37.910 --> 00:08:41.510
Because we can just rearrange
the order in which we multiply.

00:08:41.510 --> 00:08:43.520
And then how many of these
combinations are there?

00:08:43.520 --> 00:08:46.480
If we are, once again, the God
of probability, and out of six

00:08:46.480 --> 00:08:49.950
flips we're picking four--
we're choosing four that

00:08:49.950 --> 00:08:50.860
are going to end up heads?

00:08:50.860 --> 00:08:53.600
How many ways can I
pick a group of four?

00:08:53.600 --> 00:08:58.220
Well, once again, that's
times 6 choose 4.

00:08:58.220 --> 00:08:59.320
We figured out what that is.

00:08:59.320 --> 00:09:01.290
6 choose 4 is 15.

00:09:01.290 --> 00:09:13.365
So this equals 15 times .8 to
the fourth times .2 squared.

00:09:16.520 --> 00:09:24.430
And this is the probability
of four out of six heads,

00:09:24.430 --> 00:09:26.990
given an unfair coin.

00:09:26.990 --> 00:09:29.740
So what's the total
probability of getting

00:09:29.740 --> 00:09:31.430
four out of six heads?

00:09:31.430 --> 00:09:35.820
Well, it's going to be the
probability of getting the fair

00:09:35.820 --> 00:09:39.700
coin-- which is 1/3-- times the
probability of getting four out

00:09:39.700 --> 00:09:47.540
of six heads, given the fair
coin-- and that's this 15/64.

00:09:47.540 --> 00:09:52.260
Plus the probability of getting
an unfair coin-- 2/3-- times

00:09:52.260 --> 00:09:55.610
the probability of getting four
out of six heads, given the

00:09:55.610 --> 00:09:58.550
unfair coin-- and that's
what we figured out here.

00:09:58.550 --> 00:10:07.562
Times 15 times .8 to the
fourth, times .2 squared.

00:10:07.562 --> 00:10:09.950
And this is the probability of
getting four out of six heads.

00:10:09.950 --> 00:10:13.580
And let's figure
out what that is.

00:10:13.580 --> 00:10:14.910
This'll cancel out with this.

00:10:14.910 --> 00:10:17.450
This becomes 5 out of
64, that's easy enough.

00:10:17.450 --> 00:10:20.470
2/3 times 15, that's 10.

00:10:20.470 --> 00:10:23.540
And now we just have to
figure out what that is.

00:10:23.540 --> 00:10:25.900
Let's see, I'm going to go over
the time limit to see if being

00:10:25.900 --> 00:10:28.980
a Youtube partner allows me
to go over the time limit.

00:10:28.980 --> 00:10:40.370
.8 times .8 times .8 times
.8 is equal to-- and

00:10:40.370 --> 00:10:41.950
then times .2 squared.

00:10:41.950 --> 00:10:48.120
So times .2 times .2
is equal to .016.

00:10:48.120 --> 00:10:49.150
So that's that.

00:10:49.150 --> 00:10:50.580
And we said times 10, right?

00:10:50.580 --> 00:10:52.460
Because 2/3 times 15.

00:10:52.460 --> 00:10:59.950
So, times 10, is
equal to 16.384%.

00:10:59.950 --> 00:11:04.630
So the probability is-- so this
term right here-- let me write

00:11:04.630 --> 00:11:12.140
that down, and I'll switch
colors again-- this is .16384.

00:11:12.140 --> 00:11:16.040
And we're going to add
that to 5 divided by 64.

00:11:16.040 --> 00:11:17.290
So let's see.

00:11:17.290 --> 00:11:24.460
5 divided by 64 is equal to
.07, whatever, whatever.

00:11:24.460 --> 00:11:32.770
Plus .16384 is
equal to .241965.

00:11:32.770 --> 00:11:35.860
So that's the probability-- not
knowing which coin I picked

00:11:35.860 --> 00:11:39.250
out-- that's the probability of
getting four out of six heads.

00:11:39.250 --> 00:11:40.230
When you combine them you know?

00:11:40.230 --> 00:11:44.310
It could be 1/3 chance
fair, 2/3 chance unfair.

00:11:44.310 --> 00:11:50.440
So that's 24.19-- I'm keeping
the precision, just because it

00:11:50.440 --> 00:11:52.430
might come in useful
later-- % chance.

00:11:52.430 --> 00:11:55.310
So that's the probability of b.

00:11:55.310 --> 00:11:57.730
So let's see if we can clean
this up a little bit, just

00:11:57.730 --> 00:12:00.990
because I don't think we need
all of this writing now.

00:12:00.990 --> 00:12:04.050
I think we're ready to
substitute into the Bayes'

00:12:04.050 --> 00:12:08.640
formula, which we-- Bayes'
Theorem-- that we rederived.

00:12:08.640 --> 00:12:09.980
Nope, that's not what
I wanted to do.

00:12:13.110 --> 00:12:15.370
Recording longer videos is
dangerous because if I

00:12:15.370 --> 00:12:20.130
make a mistake that's
more time wasted.

00:12:20.130 --> 00:12:22.820
I don't want to delete anything
that could be useful.

00:12:25.770 --> 00:12:26.750
OK.

00:12:26.750 --> 00:12:31.650
So let's see if we can solve
the probability that we picked

00:12:31.650 --> 00:12:34.400
a fair coin, given that we
got four out of six heads.

00:12:34.400 --> 00:12:37.880
So that is going to be equal
to-- by Bayes' Theorem, which

00:12:37.880 --> 00:12:41.620
should make some sense to
you-- that is equal to the

00:12:41.620 --> 00:12:42.820
probability of b given a.

00:12:42.820 --> 00:12:48.040
So it's the probability that we
get four out of six heads,

00:12:48.040 --> 00:12:54.060
given a fair coin, times the
probability of a fair coin,

00:12:54.060 --> 00:12:56.660
over the probability of
getting four out of

00:12:56.660 --> 00:13:00.090
six heads either way.

00:13:00.090 --> 00:13:02.480
So four out of six heads, given
a fair coin-- we figured that

00:13:02.480 --> 00:13:04.270
over here-- that's 15/64.

00:13:04.270 --> 00:13:07.540
So this equals 15/64.

00:13:07.540 --> 00:13:09.240
What's the probability that
we picked a fair coin?

00:13:09.240 --> 00:13:12.110
Well, there's 15 coins and 5
of them are fair, so it's

00:13:12.110 --> 00:13:13.860
5 out of 15, so it's 1/3.

00:13:13.860 --> 00:13:16.040
So times 1/3.

00:13:16.040 --> 00:13:18.110
And what's the probability
that, in general, we picked

00:13:18.110 --> 00:13:19.250
four out of six heads?

00:13:19.250 --> 00:13:26.115
Well that's this
number: .241965.

00:13:26.115 --> 00:13:33.720
So this equals-- let's see,
this is equal to 5/64 divided

00:13:33.720 --> 00:13:38.690
by .241965, and what
does that equal to?

00:13:41.520 --> 00:13:58.380
That's 5 divided by 64 divided
by .241965 is equal to 32.--

00:13:58.380 --> 00:14:05.170
well, roughly 3%--
is equal to 32.3%.

00:14:05.170 --> 00:14:06.790
So that's amazing.

00:14:06.790 --> 00:14:09.290
Or relatively amazing.

00:14:09.290 --> 00:14:12.970
It's a little bit less than a
1/3 shot that we picked the

00:14:12.970 --> 00:14:16.730
fair coin, given that we
got four out of six heads.

00:14:16.730 --> 00:14:19.480
And what's interesting is, the
four out of six heads, it kind

00:14:19.480 --> 00:14:22.880
of decreased the probability
that we got a fair coin.

00:14:22.880 --> 00:14:26.800
Because before having any data
on what happens when we flip

00:14:26.800 --> 00:14:28.560
it, we would have had
a 1/3 probability.

00:14:28.560 --> 00:14:31.130
Which is 33.3, right?

00:14:31.130 --> 00:14:36.860
But given that we got more
heads than tails, the universal

00:14:36.860 --> 00:14:40.280
probability is telling us that,
well, if you got more heads

00:14:40.280 --> 00:14:43.340
then tails, that makes it a
little bit more likely that you

00:14:43.340 --> 00:14:45.220
picked the unfair coin, which
is a little bit more

00:14:45.220 --> 00:14:46.090
weighted to heads.

00:14:46.090 --> 00:14:48.240
But it's saying it's not that
much more likely, because this

00:14:48.240 --> 00:14:51.870
isn't that unusual of a result
to get, even with a fair coin.

00:14:51.870 --> 00:14:54.230
So that's why it became a
little bit less likely

00:14:54.230 --> 00:14:57.380
to get a fair coin.

00:14:57.380 --> 00:15:01.420
And let me give you a bit
of an intuition, visually,

00:15:01.420 --> 00:15:03.840
kind of with Set Theory,
on why that makes sense.

00:15:10.850 --> 00:15:15.450
So if we go back to Bayes'
Theorem-- let's just say

00:15:15.450 --> 00:15:22.180
that this is the universe
of all of the events.

00:15:22.180 --> 00:15:24.060
That's all of the universe.

00:15:24.060 --> 00:15:27.490
There's roughly a 1/3 chance
that I picked a fair coin.

00:15:27.490 --> 00:15:30.320
So roughly 1/3 of
this will be fair.

00:15:30.320 --> 00:15:34.040
This is fair, this is unfair.

00:15:34.040 --> 00:15:36.590
And then if I picked a fair
coin, we figured out there's

00:15:36.590 --> 00:15:41.830
roughly a 15 out of 64
shot that I get four

00:15:41.830 --> 00:15:42.460
out of six heads.

00:15:42.460 --> 00:15:44.732
So maybe that's this little
section of this-- let me do

00:15:44.732 --> 00:15:45.470
it in a different color.

00:15:45.470 --> 00:15:47.250
That's this section.

00:15:47.250 --> 00:15:50.200
And then we figured out if we
have an unfair coin-- I forgot

00:15:50.200 --> 00:15:52.590
what the exact number is-- but
there was some probability that

00:15:52.590 --> 00:15:54.720
we get four out of six heads.

00:15:54.720 --> 00:15:58.160
It's actually a little bit
bigger, it's like that.

00:15:58.160 --> 00:16:00.090
So this is getting four out
of six heads, given you

00:16:00.090 --> 00:16:01.230
got an unfair coin.

00:16:01.230 --> 00:16:02.940
This is getting four out
of six heads, given that

00:16:02.940 --> 00:16:04.290
you got a fair coin.

00:16:04.290 --> 00:16:06.680
And then this whole area is
the probability that you

00:16:06.680 --> 00:16:08.110
get four out of six heads.

00:16:08.110 --> 00:16:11.530
So all Bayes' Theorem
told us is, look, we got

00:16:11.530 --> 00:16:12.470
four out of six heads.

00:16:12.470 --> 00:16:17.590
So we're in this universe where
we got four out of six heads.

00:16:17.590 --> 00:16:21.570
And if we got four out of six
heads, 1/3 of this universe--

00:16:21.570 --> 00:16:25.850
roughly, or 32.3% of this
subset of four out of six

00:16:25.850 --> 00:16:30.620
heads-- intersects with
the fair coin universe.

00:16:30.620 --> 00:16:38.370
So this 32.3% is essentially
this fraction of the total

00:16:38.370 --> 00:16:41.740
probability of getting
four out of six heads.

00:16:41.740 --> 00:16:44.240
Anyway, hopefully that gave you
a little bit of intuition.

00:16:44.240 --> 00:16:47.170
And I hope that Youtube lets me
publish this video, because

00:16:47.170 --> 00:16:49.100
I'm on my 17th minute.

00:16:49.100 --> 00:16:51.370
I'll see you in the next video.


Probability and combinations (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:01.370
Welcome back.

00:00:01.370 --> 00:00:03.500
I actually recorded this video
earlier today, but then I

00:00:03.500 --> 00:00:05.280
realized my microphone
wasn't plugged in.

00:00:05.280 --> 00:00:08.580
And I won't name names in
terms of who unplugged it.

00:00:08.580 --> 00:00:11.260
But anyway, back
to probability.

00:00:11.260 --> 00:00:14.160
My wife is giggling
mischievously.

00:00:14.160 --> 00:00:16.910
Anyway, so let's do a
slightly harder problem

00:00:16.910 --> 00:00:17.650
than we did before.

00:00:17.650 --> 00:00:20.230
We were dealing with fair
coins, let's deal with

00:00:20.230 --> 00:00:22.100
a slightly unfair coin.

00:00:22.100 --> 00:00:24.710
Let's say I have a coin and
it's-- actually instead of

00:00:24.710 --> 00:00:26.330
unfair coin let's
do basketball.

00:00:26.330 --> 00:00:28.960
Let's say I'm shooting free
throws and I have a free

00:00:28.960 --> 00:00:35.180
throw percentage of 80%.

00:00:35.180 --> 00:00:38.260
So when I shot a free throw,
8 out of 10 times, or 80%

00:00:38.260 --> 00:00:40.610
of the time I will make it.

00:00:40.610 --> 00:00:43.890
But that also says that 20%
of time I will miss it.

00:00:43.890 --> 00:00:47.580
So given that, if I were to
take-- I don't know-- 5 free

00:00:47.580 --> 00:00:54.590
throws, what is the probability
that I make at least 3

00:00:54.590 --> 00:00:56.520
of the 5 free throws?

00:01:00.030 --> 00:01:03.680
Well, let's think of it this
way, what is the probability

00:01:03.680 --> 00:01:07.840
of any particular combination
of making 3 out of the 5?

00:01:07.840 --> 00:01:09.560
So what do I mean by that?

00:01:09.560 --> 00:01:11.370
Let me pick a particular
combination.

00:01:11.370 --> 00:01:16.580
Let's say it's a basket,
basket, basket, and

00:01:16.580 --> 00:01:18.870
then I miss, miss.

00:01:18.870 --> 00:01:20.990
So that would be I
made 3 out of the 5.

00:01:20.990 --> 00:01:26.490
It could be-- I don't know--
basket, miss, basket,

00:01:26.490 --> 00:01:28.570
miss, basket.

00:01:28.570 --> 00:01:30.760
And there's a bunch of them and
we'll actually try to figure

00:01:30.760 --> 00:01:32.540
out how many of them there are.

00:01:32.540 --> 00:01:38.060
But what is the probability of
this particular combination?

00:01:38.060 --> 00:01:41.950
Well, I have an 80% chance of
making this first basket.

00:01:41.950 --> 00:01:43.210
Times 80%.

00:01:43.210 --> 00:01:45.320
That's a times right there.

00:01:45.320 --> 00:01:48.670
Times 80%.

00:01:48.670 --> 00:01:50.420
And then what's my
probability of missing?

00:01:50.420 --> 00:01:51.680
Well, that's 20%, right?

00:01:51.680 --> 00:01:53.460
Times 0.2.

00:01:53.460 --> 00:01:54.790
times 0.2.

00:01:54.790 --> 00:02:03.170
So this sequels 0.8 to the
third power times 0.2 squared.

00:02:03.170 --> 00:02:05.480
What's the probability of
getting this exact combination?

00:02:05.480 --> 00:02:08.650
Well, it's 0.8
times-- then I miss.

00:02:08.650 --> 00:02:10.110
There's a 20% chance of that.

00:02:10.110 --> 00:02:17.820
So times 0.2 times 0.8
times 0.2 times 0.8.

00:02:17.820 --> 00:02:20.190
We could rearrange this because
when you multiply numbers it

00:02:20.190 --> 00:02:23.450
doesn't matter what order
you multiply them in.

00:02:23.450 --> 00:02:26.280
So this is the same thing
as 0.8 times 0.8 times

00:02:26.280 --> 00:02:28.410
0.8 times 0.2 times 0.2.

00:02:28.410 --> 00:02:32.680
So this is also the same
thing as 0.8 to the

00:02:32.680 --> 00:02:36.836
third times 0.2 squared.

00:02:39.790 --> 00:02:44.340
The probability of getting any
particular combination of 3

00:02:44.340 --> 00:02:48.470
baskets and 2 misses is going
to be 0.8 to third

00:02:48.470 --> 00:02:51.470
times 0.2 squared.

00:02:51.470 --> 00:02:52.750
Now what's the total
probability of

00:02:52.750 --> 00:02:53.340
getting 3 out of 5?

00:02:53.340 --> 00:02:55.720
Well, it's going to be the sum
of all of these combinations.

00:02:55.720 --> 00:02:58.380
You know, I could list them
all, but we hopefully now

00:02:58.380 --> 00:03:02.520
are proficient enough in
combinatorics and combinations

00:03:02.520 --> 00:03:05.470
to figure out how many
different ways, if we have 5

00:03:05.470 --> 00:03:08.950
baskets and we're picking-- or
we have 5 shots and we're

00:03:08.950 --> 00:03:11.530
picking 3 of them to be the
ones that are basket shots.

00:03:11.530 --> 00:03:12.180
So what do I mean?

00:03:12.180 --> 00:03:21.670
So let's say my 5 shots-- you
know, I've shot 1, 2, 3, 4, 5.

00:03:21.670 --> 00:03:23.810
Out of these five, I'm
going to choose 3.

00:03:23.810 --> 00:03:26.140
So once again, I'm putting my
hat on as the god of

00:03:26.140 --> 00:03:30.850
probability and I will choose 3
of these shots to be the ones

00:03:30.850 --> 00:03:35.400
that happen to be the
ones that get made.

00:03:35.400 --> 00:03:41.030
So essentially, out of
5 I am choosing 3.

00:03:41.030 --> 00:03:43.400
5 choose 3.

00:03:43.400 --> 00:03:44.630
And what does that equal to?

00:03:44.630 --> 00:03:52.520
That's 5 factorial over 3
factorial times 5 minus 3

00:03:52.520 --> 00:03:55.850
factorial, so that's
2 factorial.

00:03:55.850 --> 00:04:04.980
And that equals 5 times 4 times
3 times 2 times 1 over 3 times

00:04:04.980 --> 00:04:07.650
2 times 1 times 2 times 1.

00:04:07.650 --> 00:04:08.725
We can ignore all the 1's.

00:04:08.725 --> 00:04:09.220
Let's see.

00:04:09.220 --> 00:04:10.500
We get 3 times 2 times 1.

00:04:10.500 --> 00:04:11.480
3 times 2 times 1.

00:04:11.480 --> 00:04:13.490
We can cancel that.

00:04:13.490 --> 00:04:16.720
This 1 we can ignore, and
then this 2 and then

00:04:16.720 --> 00:04:17.960
this turns into 2.

00:04:17.960 --> 00:04:20.130
So there are 10
possible combinations.

00:04:20.130 --> 00:04:20.980
These are two of them.

00:04:20.980 --> 00:04:23.710
You know, basket, basket,
basket, miss, miss.

00:04:23.710 --> 00:04:25.660
Basket, miss, basket,
miss, basket.

00:04:25.660 --> 00:04:27.760
And you know, it's a good
exercise for you to list

00:04:27.760 --> 00:04:28.760
the other 8 of them.

00:04:28.760 --> 00:04:32.650
But using just the binomial
coefficient, and hopefully you

00:04:32.650 --> 00:04:36.000
have an intuition of why that
works and I'd be happy to make

00:04:36.000 --> 00:04:38.690
more videos if you feel that
that you need more explanation.

00:04:38.690 --> 00:04:40.220
But I made a couple.

00:04:40.220 --> 00:04:42.050
There are 10 combinations.

00:04:42.050 --> 00:04:45.220
So essentially, the probability
of getting exactly 3 out of 5

00:04:45.220 --> 00:04:50.970
baskets, if I am an 80% free
throw shot, is going to

00:04:50.970 --> 00:04:54.030
be-- switch colors.

00:04:54.030 --> 00:05:01.360
The probability of 3 out of 5
baskets is going to be equal to

00:05:01.360 --> 00:05:06.720
the probability of each of the
combinations, which is 0.8 to

00:05:06.720 --> 00:05:12.660
the third times 0.2 squared.

00:05:12.660 --> 00:05:15.020
I make 3, miss 2.

00:05:15.020 --> 00:05:17.150
And then, times the total
number of combinations.

00:05:17.150 --> 00:05:19.690
Each of these has a
probability of this much.

00:05:19.690 --> 00:05:22.650
And then there's 10 different
arrangements that I could make.

00:05:22.650 --> 00:05:26.050
There's 10 different ways of
getting 3 baskets and 2 misses.

00:05:26.050 --> 00:05:29.900
So times 10, and what
does that equal to?

00:05:29.900 --> 00:05:36.320
Let me get my high-end
calculator here.

00:05:36.320 --> 00:05:38.800
So let's see what that is.

00:05:38.800 --> 00:05:53.860
That is 0.8 times 0.8 times 0.8
times 0.2 times 0.2 times 10.

00:05:53.860 --> 00:05:56.450
Equals 20.48.

00:05:56.450 --> 00:06:03.980
So it's essentially, a 20.48%
chance that I get exactly

00:06:03.980 --> 00:06:06.076
3 out of 5 of the baskets.

00:06:06.076 --> 00:06:10.340
Now let's make it slightly
more interesting.

00:06:10.340 --> 00:06:12.980
Let's say I don't want it as a
probability of 3 out of the 5.

00:06:12.980 --> 00:06:15.050
And this is actually something
that probably, people

00:06:15.050 --> 00:06:16.145
are more likely to ask.

00:06:19.080 --> 00:06:25.260
What is the probability of
getting at least 3 baskets?

00:06:28.350 --> 00:06:30.070
Well, if you think about it,
this is the probability.

00:06:30.070 --> 00:06:33.490
This is equal to the
probability of getting 3 out of

00:06:33.490 --> 00:06:38.880
5 baskets, plus the probability
of getting exactly 4 out of 5

00:06:38.880 --> 00:06:42.380
baskets, plus the probability
of getting exactly 5

00:06:42.380 --> 00:06:43.992
out of 5 baskets.

00:06:43.992 --> 00:06:45.720
We already figured
this one out.

00:06:45.720 --> 00:06:47.390
That's 20.48%.

00:06:47.390 --> 00:06:50.170
So what's the probability of
getting 4 out of 5 baskets?

00:06:50.170 --> 00:06:55.030
Well, once again, if we want
exactly 4 out of 5 baskets,

00:06:55.030 --> 00:07:01.320
so an example could be-- I
don't know-- miss, basket,

00:07:01.320 --> 00:07:03.810
basket, basket, basket.

00:07:03.810 --> 00:07:06.010
What's the probability of
any one of the combinations

00:07:06.010 --> 00:07:07.180
where I make 4 baskets?

00:07:07.180 --> 00:07:14.170
Well, it's going to be 0.8 to
the fourth times-- and then I

00:07:14.170 --> 00:07:16.730
have a 20% chance
that one miss.

00:07:16.730 --> 00:07:19.120
You know, it could have
been basket, miss,

00:07:19.120 --> 00:07:22.280
basket, basket, basket.

00:07:22.280 --> 00:07:25.560
That's also exactly 4, but
when you multiply them, the

00:07:25.560 --> 00:07:28.590
probability of getting any
one of these particular

00:07:28.590 --> 00:07:31.300
combinations is exactly this--
0.8 to the fourth times 0.2.

00:07:33.980 --> 00:07:38.190
If I have 5 baskets, how many
ways can I pick 4 of them to be

00:07:38.190 --> 00:07:39.870
the ones that I make if I'm
once again the god

00:07:39.870 --> 00:07:41.460
of probability?

00:07:41.460 --> 00:07:51.540
So this is going to be 0.8 to
the fourth times 0.2 times 0.2

00:07:51.540 --> 00:07:56.330
times-- out of 5 baskets, I'm
choosing 4 that I'm

00:07:56.330 --> 00:07:57.060
going to make.

00:07:57.060 --> 00:07:59.530
So this is the number of
combinations where I

00:07:59.530 --> 00:08:00.870
get 4 out of the 5.

00:08:00.870 --> 00:08:02.600
So what is 5 choose 4?

00:08:02.600 --> 00:08:09.210
That's 5 factorial over 4
factorial times 1 factorial.

00:08:09.210 --> 00:08:10.995
Well, that equals just 5.

00:08:10.995 --> 00:08:13.520
You can work that out.

00:08:13.520 --> 00:08:16.540
So let's just figure this out.

00:08:16.540 --> 00:08:24.340
This is going to be 0.8
times 0.8 times 0.8--

00:08:24.340 --> 00:08:28.770
that's 3-- times 0.8.

00:08:28.770 --> 00:08:31.430
That equals-- did
I do that right?

00:08:31.430 --> 00:08:32.476
Let's see.

00:08:32.476 --> 00:08:33.310
0.1.

00:08:33.310 --> 00:08:36.600
Wait.

00:08:36.600 --> 00:08:41.280
0.8 times 0.8--
yeah, that's right.

00:08:41.280 --> 00:08:45.690
Times 0.2 times 5.

00:08:51.510 --> 00:08:53.310
So 40.96%.

00:08:53.310 --> 00:08:57.670
So this is 40.96%.

00:08:57.670 --> 00:09:01.500
So roughly, 41% chance that I
get exactly 4 out of 5 baskets.

00:09:01.500 --> 00:09:03.870
Which is interesting because
that's kind of my free

00:09:03.870 --> 00:09:04.630
throw percentage.

00:09:04.630 --> 00:09:08.200
So it's almost a little less--
you know, 2/3 shot of kind of

00:09:08.200 --> 00:09:09.660
hitting my free throw
percentage on the

00:09:09.660 --> 00:09:11.110
mark on that time.

00:09:11.110 --> 00:09:13.020
And what's the probability
of getting 5 out of 5

00:09:13.020 --> 00:09:14.760
Well, there's only one way
of getting 5 out of 5.

00:09:14.760 --> 00:09:16.550
You have to get all 5 of them.

00:09:16.550 --> 00:09:21.110
So this is 0.8 to
the fifth power.

00:09:21.110 --> 00:09:24.110
Let me get the calculator back.

00:09:24.110 --> 00:09:35.640
So it's 0.8 times 0.-- oh,
wait-- times 0.8 times 0.8

00:09:35.640 --> 00:09:46.220
times 0.8 times 0.8
equals 0.3276.

00:09:46.220 --> 00:09:50.300
So 32.77% shot.

00:09:50.300 --> 00:09:52.330
And then we can add them
all up because we want the

00:09:52.330 --> 00:09:54.130
probability of at least 3.

00:09:54.130 --> 00:09:56.610
So it's going to be that, the
probability of getting 5 out of

00:09:56.610 --> 00:09:59.990
5, plus the probability of
getting 4 out of 5,

00:09:59.990 --> 00:10:05.660
which is 0.4096.

00:10:05.660 --> 00:10:09.100
Plus, the probability
of getting 3 out of 5.

00:10:09.100 --> 00:10:16.830
So that's 0.2048
equals 0.94208.

00:10:16.830 --> 00:10:22.520
So 94.21-- roughly, rounding--
% chance, which makes sense.

00:10:22.520 --> 00:10:26.900
If I have an 80% free throw
percentage on any one shot, I

00:10:26.900 --> 00:10:30.750
have a very high probability of
getting at least 3 out of 5

00:10:30.750 --> 00:10:32.210
when I go to the
free throw line.

00:10:32.210 --> 00:10:33.590
Anyway, I'm all out of time.

00:10:33.590 --> 00:10:35.540
I'll see you in the next video.


Probability using combinations

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.730
So you might be wondering why I
went off into permutations and

00:00:03.730 --> 00:00:06.550
combinations in the probability
playlist, and I think you'll

00:00:06.550 --> 00:00:08.560
learn in this video.

00:00:08.560 --> 00:00:11.626
So let's say I want to figure
out the probability-- I'm going

00:00:11.626 --> 00:00:14.200
to flip a coin eight times
and it's a fair coin.

00:00:14.200 --> 00:00:16.710
And I want to figure out
the probability of getting

00:00:16.710 --> 00:00:20.480
exactly 3 out of 8 heads.

00:00:20.480 --> 00:00:23.740
So I say 3/8 heads, but 3 of my
flips are going to be heads and

00:00:23.740 --> 00:00:26.050
the rest are going to be tails.

00:00:26.050 --> 00:00:27.720
So how do I think about that?

00:00:27.720 --> 00:00:31.460
Well, let's go back to one
of the early definitions

00:00:31.460 --> 00:00:32.910
we used for probability.

00:00:32.910 --> 00:00:37.810
And that says, the probability
of anything happening is the

00:00:37.810 --> 00:00:42.300
probability of the number of
equally probable events into

00:00:42.300 --> 00:00:45.040
which what we're
stating is true.

00:00:45.040 --> 00:00:50.590
So in which the number of
events-- I guess trials or

00:00:50.590 --> 00:00:55.390
situations-- in which we
get 3 heads, and exactly 3

00:00:55.390 --> 00:00:57.200
heads, we're not saying
greater than 3 heads.

00:00:57.200 --> 00:00:59.770
So 4 heads won't count and 2
heads won't count, 5 heads

00:00:59.770 --> 00:01:02.480
won't-- only 3 heads.

00:01:02.480 --> 00:01:10.320
And then, over the total number
of equally probable trials--

00:01:10.320 --> 00:01:13.476
not trials, total number of
equally possible outcomes.

00:01:13.476 --> 00:01:15.880
I should be using
the word outcomes.

00:01:15.880 --> 00:01:18.640
So just with the word outcomes
it should be the total number

00:01:18.640 --> 00:01:21.070
of outcomes in which what
we're saying happens.

00:01:21.070 --> 00:01:26.050
So we get 3 heads over the
total possible outcomes.

00:01:26.050 --> 00:01:27.705
So let's do the
bottom part first.

00:01:31.620 --> 00:01:35.210
What are the total possible
outcomes if I'm flipping

00:01:35.210 --> 00:01:38.070
a fair coin eight times?

00:01:38.070 --> 00:01:39.980
Well, the first time I flip
it I either get heads or

00:01:39.980 --> 00:01:42.620
tails, so I get 2 outcomes.

00:01:42.620 --> 00:01:45.230
And then when I flip it
again I get 2 more come

00:01:45.230 --> 00:01:45.970
for the second one.

00:01:45.970 --> 00:01:47.810
And then, how many
total outcomes?

00:01:47.810 --> 00:01:49.790
Well, that's 2 times 2 because
I could have got 2 in the

00:01:49.790 --> 00:01:51.570
first, 2 in the second flip.

00:01:51.570 --> 00:01:53.560
And then essentially we
would multiply 2 times

00:01:53.560 --> 00:01:55.210
the number of flips.

00:01:55.210 --> 00:02:00.280
So that's 5, 6, 7, 8, and
that equals 2 to the eighth.

00:02:00.280 --> 00:02:03.400
So the number of outcomes is
just going to be 2 to the

00:02:03.400 --> 00:02:04.860
total number of flips.

00:02:04.860 --> 00:02:07.020
And hopefully that
make sense to you.

00:02:07.020 --> 00:02:08.400
If not, you might want
to re-watch some of

00:02:08.400 --> 00:02:09.440
the earlier videos.

00:02:09.440 --> 00:02:11.280
But that's the easy part.

00:02:11.280 --> 00:02:14.690
So there's 2 to the eighth
possible outcomes when you

00:02:14.690 --> 00:02:17.340
flip a fair coin eight times.

00:02:17.340 --> 00:02:20.060
So how many of those outcomes
are going to result

00:02:20.060 --> 00:02:23.761
in exactly 3 heads?

00:02:23.761 --> 00:02:25.200
Let's think of it this way.

00:02:25.200 --> 00:02:29.980
Let's give a name to
each of our flips.

00:02:29.980 --> 00:02:31.340
Let's give a name to them.

00:02:31.340 --> 00:02:35.260
So let me make a little column,
we'll call these the flips.

00:02:35.260 --> 00:02:37.890
This is my flips column.

00:02:37.890 --> 00:02:39.020
And I could name them anything.

00:02:39.020 --> 00:02:40.280
I could name them
Larry, Curly, Moe.

00:02:40.280 --> 00:02:43.010
I could name them-- well, I
would need 5 more names for

00:02:43.010 --> 00:02:46.840
them, but I could name them the
7 dwarfs or the 8 dwarfs really

00:02:46.840 --> 00:02:50.000
because I have 8 flips.

00:02:50.000 --> 00:02:50.860
I'll number the flips.

00:02:50.860 --> 00:02:58.530
Flip 1, 2, 3, 4, 5, 6, 7, 8.

00:02:58.530 --> 00:03:00.270
And I'm the god of probability.

00:03:00.270 --> 00:03:04.260
And essentially, I need to just
pick 3 of these flips that

00:03:04.260 --> 00:03:05.890
are going to result in heads.

00:03:05.890 --> 00:03:09.340
So another way to think about
it is, these could be 8 people

00:03:09.340 --> 00:03:13.680
and I could pick which of
these-- how many ways can I

00:03:13.680 --> 00:03:16.710
pick 3 of these people
to put into the car?

00:03:16.710 --> 00:03:19.760
How many ways can I pick 3 of
these people to sit in chairs.

00:03:19.760 --> 00:03:22.510
And it doesn't matter the
order that I pick them in.

00:03:22.510 --> 00:03:25.770
It doesn't matter if I say the
people that are going to get in

00:03:25.770 --> 00:03:28.280
the car are going to
be people 1, 2, 3 3.

00:03:28.280 --> 00:03:30.970
Or if I say 3, 2, and 1,
or if I say 2, 3, and 1.

00:03:30.970 --> 00:03:35.030
Those are all the
same combination.

00:03:35.030 --> 00:03:37.290
So similarly, if I'm just
picking flips and I have

00:03:37.290 --> 00:03:41.070
to say, OK, 3 of these
flips are going to get

00:03:41.070 --> 00:03:43.830
into the heads car.

00:03:43.830 --> 00:03:45.830
Heads is like they're sitting,
they're people sitting down.

00:03:45.830 --> 00:03:47.040
I don't want to
confuse you too much.

00:03:47.040 --> 00:03:49.320
But essentially I'm
just going to choose 3

00:03:49.320 --> 00:03:51.340
things out of the 8.

00:03:51.340 --> 00:03:55.580
So I'm essentially just saying,
how many combinations can I get

00:03:55.580 --> 00:03:57.810
where I pick 3 out of these 8.

00:03:57.810 --> 00:04:00.800
And so that should immediately
ring a bell that we're

00:04:00.800 --> 00:04:05.290
essentially saying, out of 8
things we're going to choose 3.

00:04:08.440 --> 00:04:11.650
How many combinations of 3
can we pick of 8 and that we

00:04:11.650 --> 00:04:13.590
went over in the last video.

00:04:13.590 --> 00:04:15.500
And let's do it with
the formula first.

00:04:15.500 --> 00:04:17.950
So let me write the formula up
here just so you remember it,

00:04:17.950 --> 00:04:20.230
but I also want to give you the
intuition again,

00:04:20.230 --> 00:04:21.510
for the formula.

00:04:21.510 --> 00:04:28.430
So in general, we said, n
choose k, that is equal to n

00:04:28.430 --> 00:04:37.130
factorial over k factorial
times n minus k factorial.

00:04:37.130 --> 00:04:43.640
So in this situation that
would equal 8 factorial over

00:04:43.640 --> 00:04:46.360
3 factorial times what?

00:04:46.360 --> 00:04:49.510
8 minus k-- times 5 factorial.

00:04:49.510 --> 00:04:53.790
Or another way of writing this,
this would be 8 times 7 times 6

00:04:53.790 --> 00:04:58.880
times 5 times 4 times 3 times 2
times 1 over-- I'll just write

00:04:58.880 --> 00:05:01.050
3 factorial for a second.

00:05:01.050 --> 00:05:05.520
Then times 5 times 4
times 3 times 2 times 1.

00:05:05.520 --> 00:05:08.440
And of course, that and that
cancel out and all you're

00:05:08.440 --> 00:05:13.400
left with is 8 times 7
time 6 over 3 factorial.

00:05:13.400 --> 00:05:16.000
And I did this for reason
because I want you to re-get

00:05:16.000 --> 00:05:24.100
the intuition at least for
this part of the formula.

00:05:24.100 --> 00:05:27.790
That's essentially just saying,
how many permutations can I--

00:05:27.790 --> 00:05:30.210
how many ways can I pick
3 things out of 8?

00:05:30.210 --> 00:05:32.420
And that's essentially saying,
well, before I pick anything

00:05:32.420 --> 00:05:33.860
I could pick 1 of 8.

00:05:33.860 --> 00:05:37.230
Then I have 7 left to pick
from for the second spot.

00:05:37.230 --> 00:05:40.650
And then I have 6 left to
pick for the third spot.

00:05:40.650 --> 00:05:43.670
And so that's essentially
the number of permutations.

00:05:43.670 --> 00:05:47.910
But since we don't care what
order we picked them in, we

00:05:47.910 --> 00:05:51.350
need to divide by the number of
ways we can rearrange 3 things,

00:05:51.350 --> 00:05:54.960
and that's where the 3
factorial comes from.

00:05:54.960 --> 00:05:58.460
And so hopefully I didn't
confuse you, but if I did you

00:05:58.460 --> 00:06:01.560
can go back to this formula
for the binomial coefficient.

00:06:01.560 --> 00:06:03.090
But it's good to
have the intuition.

00:06:03.090 --> 00:06:04.910
And then once we're
at this point we can

00:06:04.910 --> 00:06:06.460
just calculate this.

00:06:06.460 --> 00:06:07.050
Well, what's this?

00:06:07.050 --> 00:06:11.720
This is 8 times 7 times
6 over 3 factorial of

00:06:11.720 --> 00:06:12.720
3 times 2 times 1.

00:06:12.720 --> 00:06:14.090
So that's 6.

00:06:14.090 --> 00:06:17.130
The 6 cancels out,
so it's 8 times 7.

00:06:17.130 --> 00:06:20.760
So there's 8 times
7, or what is that?

00:06:20.760 --> 00:06:21.060
56.

00:06:23.560 --> 00:06:24.800
That's equal to 56.

00:06:24.800 --> 00:06:29.140
So there's 56 different ways
to pick 3 things out of 8.

00:06:29.140 --> 00:06:34.180
Or if I have 8 people there's
56 ways of picking 3 people to

00:06:34.180 --> 00:06:37.030
sit in the car or however
you want t view it.

00:06:37.030 --> 00:06:40.430
But if I have 8 flips there's
56 ways of picking 3 of

00:06:40.430 --> 00:06:42.270
those flips to be heads.

00:06:42.270 --> 00:06:44.480
So let's go to our original
probability problem.

00:06:44.480 --> 00:06:47.180
What is the probably that
I get 3 out of 8 heads?

00:06:47.180 --> 00:06:53.030
Well, it's the number of ways
I can pick 3 out of those 8,

00:06:53.030 --> 00:06:58.170
so it equals 56, over the
total number of outcomes.

00:06:58.170 --> 00:07:00.430
The total number of outcomes
is 2 to the eighth.

00:07:00.430 --> 00:07:03.100
Another way I could write
that-- 56, let me unseparate.

00:07:03.100 --> 00:07:08.670
That's 8 times 7 over
2 to the eighth.

00:07:08.670 --> 00:07:11.140
8 is 2 to the third.

00:07:11.140 --> 00:07:14.660
Let me erase some of this.

00:07:14.660 --> 00:07:17.940
Not with that color.

00:07:17.940 --> 00:07:19.690
Let me erase that.

00:07:19.690 --> 00:07:23.400
Let me erase all of
this just so I space.

00:07:23.400 --> 00:07:27.050
And I will switch
colors for variety.

00:07:27.050 --> 00:07:28.750
Let me use the small pen.

00:07:28.750 --> 00:07:30.590
OK, so I'm back.

00:07:30.590 --> 00:07:33.750
All right, so 8 is the same
thing as 2 to the third times

00:07:33.750 --> 00:07:35.760
7-- this is all just
mathematical simplification,

00:07:35.760 --> 00:07:37.940
but it's useful-- over
2 to the eighth.

00:07:37.940 --> 00:07:42.430
And so, if we just divide both
sides-- the numerator and the

00:07:42.430 --> 00:07:46.075
denominator by 2 to the
third, this becomes 1.

00:07:46.075 --> 00:07:48.830
This becomes 2 to the fifth.

00:07:48.830 --> 00:07:58.460
And so it becomes 7/32.

00:07:58.460 --> 00:08:00.070
Is that right?

00:08:00.070 --> 00:08:06.160
So if I were to pick 3 out of
8-- yep, I think that is right.

00:08:06.160 --> 00:08:07.510
And so what does that
turn out to be?

00:08:07.510 --> 00:08:10.442
Let me get my calculator.

00:08:10.442 --> 00:08:10.910
[INAUDIBLE]

00:08:10.910 --> 00:08:13.260
to make careless mistakes.

00:08:13.260 --> 00:08:13.680
Let's see.

00:08:13.680 --> 00:08:15.126
My calculator seems
to have disappeared.

00:08:15.126 --> 00:08:18.140
Let me get it back.

00:08:18.140 --> 00:08:19.530
There it is.

00:08:19.530 --> 00:08:21.140
OK.

00:08:21.140 --> 00:08:32.070
7 divided by 32 is
equal to 0.21875.

00:08:32.070 --> 00:08:37.370
Which is equal to 21.-- you
know, if I were to round

00:08:37.370 --> 00:08:40.820
roughly-- 21.9% chance.

00:08:40.820 --> 00:08:44.820
So there's a little bit better
than 1 in 5 chance that I get

00:08:44.820 --> 00:08:49.420
exactly 3 out of the
8 flips as heads.

00:08:49.420 --> 00:08:51.840
Hopefully I didn't confuse you
and now you can apply that

00:08:51.840 --> 00:08:52.730
to pretty much anything.

00:08:52.730 --> 00:08:54.720
You could say, well, what is
the probability of getting-- if

00:08:54.720 --> 00:08:58.430
I flip a fair coin-- of getting
exactly 7 out of 8 heads?

00:08:58.430 --> 00:09:01.780
Or you could say, what's
the probability of getting

00:09:01.780 --> 00:09:03.470
2 out of 100 heads?

00:09:03.470 --> 00:09:07.370
And you could use it the exact
same way we did this problem.

00:09:07.370 --> 00:09:09.410
I'll see you in the next video.