Combinations

WEBVTT
Kind: captions
Language: en

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In the last video we figured
out how many different ways

00:00:04.390 --> 00:00:11.790
could 5 people sit in 3 chairs.

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So for example, if this is
chair one, this is chair

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two, this is chair three.

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We said, well, in chair one
we could put 5 people.

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No one's sitting down.

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Then there's only going to be 4
people left, so we could put 4

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different people in chair 2.

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And then there would be 3
people left that we could

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put in chair three.

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So the total permutations, the
different ways that people

00:00:34.220 --> 00:00:36.340
could sit in the different
yours if we cared about the

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order-- if we cared about which
chair they were sitting down

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in-- it would be 5
times 4 times 3.

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And another way of thinking
about that-- 5 times 4 times

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3-- that's the same thing.

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That equals 5 times 4 times 3
times 2 times 1 over what?

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Over 2 times 1.

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And that's the same thing as 5
factorial over 2 factorial.

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Now where does
this 2 come from?

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How is 2 related
to the 5 and 3?

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What's the difference
between the two?

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So this is the same thing,
this is equal to 5 factorial

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over 5 minus 3 factorial.

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And that in general is how
we figure out, how many

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permutations can 5 things
place themselves or be

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placed into 3 positions?

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And the general formula
is-- and we learned

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this in the last video.

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And I will switch colors.

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If we want to put n things into
k positions, and k has to

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be less than or equal to n.

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Well actually, it
doesn't have to.

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But for our purposes right now
we'll assume it is because our

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formula might break
down if it weren't.

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And that equals n factorial
over n minus k factorial.

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I always find this harder
to memorize than just

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thinking about the spot.

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Then just saying, oh well,
you know, 5 people, 5 of

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the things could be here.

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And then once one thing is
here, there's 4 possibilities

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left and then there's
3 possibilities left.

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The way I think of it I
take the first k terms

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of the n factorial.

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Or in this case, I take
the first three terms

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of the 5 factorial.

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5 times 4 times 3.

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That's how I think
of permutations.

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This is great if we cared--
let's say these are

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people A, B, C, D, E.

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So these are the 5 people that
are going to sit in the chairs.

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This is great if we wanted to
count the permutation ABC as

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different from the permutation
ACB, as different from the

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permutation-- I don't know--
BAC as different from

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the permutation BCA.

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Because remember, when we did
this we actually cared about

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where they're sitting.

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And in the previous video we
double counted everything

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because it matters if person
A is in seat one and

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person B is in seat two.

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And then if they switch
we recounted, right?

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This is where they switch.

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But what if we didn't
care about that?

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What if we didn't care
who's in what seat?

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We just wanted to know how
many different ways can

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the 5 people sit down?

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So we want to count all the
situations where person A, B,

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and C are sitting down as
essentially one situation.

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We don't care who's
sitting in which chair.

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We just care that those are
the 3 people sitting down.

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That's the set, the subset
of the people sitting down.

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And so the question then
becomes not how many different

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permutations or how many
different ways can the people

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sit down, the question becomes,
how many subsets of 3 can

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we take out of a set of 5?

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And I know I'm kind of jumping
up around a little bit, but

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that's essentially what
a combination is.

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A combination is a permutation
where you don't care

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about the order.

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So how do we figure it out?

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Well, when we figured out the
permutations using this formula

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we counted-- for example, we
counted ABC, ACB, BAC,

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BCA, and let's see.

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There should be two
more permutations.

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CAB and CBA.

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We counted all 6 of these
as different permutations.

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But in our combinations we're
going to want to-- this is all

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essentially the same
combination because we don't

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care about the order.

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So for any 3 different people
that are in these seats,

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there's actually going to be 6
permutations that we're

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counting when we do
the permutations.

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So if we want the combinations
we'll just divide by the number

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of ways we can rearrange
3 people into 3 seats.

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That's essentially
what we did here.

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So how many different
ways can you arrange 3

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people into 3 seats?

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Well, this is kind of another
permutation problem.

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The first seat you could put 3
different people, the second

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seat you put 2 different
people, and the last seat--

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well, there's only
1 person left.

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So it equals 3 factorial,
which is equal to 6.

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This is equal to 3 factorial,
which is equal to 6.

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Hope I'm not confusing.

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What I'm just trying to say is
when you did a permutation we

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counted all of the different
orders of how people could

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arrange themselves.

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And what I'm saying now is
well, how many different ways

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can people arrange themselves?

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Well, it's going to be the
number of places factorial

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because if we have 3 people
in 3 spots or let's say,

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4 people in 4 spots.

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The fourth spot can have 4
people, the second spot can

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have 3, and so forth, the third
spot could have 2, and the

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last spot will only have 1.

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So it's the number of spots
factorial is how many

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permutations we're counting.

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When we have just the same
people, they're just playing

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musical chairs in the
exact same seats.

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So on order to figure out the
combination, so if we wanted to

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say how many people-- let's
say if we had 5 people.

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How many different groups
of 3 can be seated?

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And we don't want to double.

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We don't want to more
than double count.

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I don't know what the
word is for counting

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something six times.

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Well, it's just going to be the
same thing as the permutation

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divided by all of the
extra counting we did.

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We'll just divide by the number
of ways that 3 people can

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arrange themselves in 3 seats.

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And that's 3 factorial.

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And I hope I'm making sense.

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Maybe I'll do a couple more
examples in other videos.

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And definitely request
it if you think this

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is extra confusing.

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So in general, if we say, what
are the different ways that

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n things can be chosen?

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Or the number of combinations
that n things can be chosen

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into sets of r, where r is
less than or equal to n.

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It's equal to the number of
permutations you could create

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of putting n things into r
spots, divided by r factorial.

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We're going to divide by the
number of ways the r spots

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themselves could be rearranged
because we don't want to

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count those as extra.

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And so if we go back to this
formula up here, well, this

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was a k, but now we're
saying it's an r.

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This is the same thing as--
so the permutations was

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n factorial over n
minus r factorial.

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And now we're dividing
everything by r factorial.

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So that equals-- let
me just write this.

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This is often written
as n choose r.

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Another way it's
written is n choose r.

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This is called the binomial
coefficient and we'll do a

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whole series of modules on that
as well because this actually

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shows up in polynomial
expansion when you take

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polynomials to powers.

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But this is equal to n
factorial over r factorial

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divided by n minus r factorial.

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You could memorize this.

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You know, it's useful if
you want to do things

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quickly on tests.

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But it's very important to
think about where it came from.

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The n factorial over the n
minus r factorial-- this

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is just the permutation.

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And what is that?

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Well, that was just the first
r-- I guess you could call

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it the first r factors.

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The r largest factors
of n factorial.

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That's all that is.

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And then when we do the
combinations we divided by r

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factorial because we want to
divide it by all the different

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arrangements that the people
could seat themselves

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in r seats.

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Or, the balls could
be placed in r cups.

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So in this situation if we want
to know how many different

00:09:03.450 --> 00:09:08.090
groups of 3 can be selected
from 5 people or from 5

00:09:08.090 --> 00:09:16.370
letters, it's going to be 5
factorial over 3 factorial

00:09:16.370 --> 00:09:21.040
times 5 minus 3 factorial.

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And that's 5 times 3 times
2 times 1, over-- 3

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factorial is just 6.

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We'll put that aside
for a second.

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Divided by-- this
is 3 factorial.

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2 times 1.

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So notice, this is a
permutation part right here.

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This term just gets rid of
the two lowest factors.

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You get 5 times 3.

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Oh, sorry there's a 4.

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5 times 4 times 3, which is
the number of permutations.

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And then we divide by 6 because
we get 6 permutations for

00:09:52.330 --> 00:09:55.150
really every combination.

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Maybe that confused you.

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But anyway, so we get 5 times
4 times 3 divided by 6.

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And that's what?

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5 times 12 divided by 6,
which is equal to 5 times 2.

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There's 10 possible ways
that we can take sets of 3

00:10:11.330 --> 00:10:13.750
from a group of 5 things.

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See you in the next video.


Permutations

WEBVTT
Kind: captions
Language: en

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Let's say I have 3 chairs.

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They'll just be
blanks: 1, 2, and 3.

00:00:09.210 --> 00:00:12.210
3 chairs, and I have 7 people.

00:00:12.210 --> 00:00:18.310
I'll call them people A, B,
C, D, E-- how many is that?

00:00:18.310 --> 00:00:20.630
That's 5-- F, G.

00:00:20.630 --> 00:00:23.420
So we have 7 people and what
I want to know is how many

00:00:23.420 --> 00:00:28.400
different ways can these 7
people sit in these 3 chairs?

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Well, before anyone sits down
if we just pick a chair, these

00:00:32.850 --> 00:00:35.620
are just arbitrary labels for
the chairs, but let's start at

00:00:35.620 --> 00:00:37.310
the left with chair number one.

00:00:37.310 --> 00:00:40.760
If no one is sitting down, how
many different people could

00:00:40.760 --> 00:00:42.130
sit down in chair one?

00:00:42.130 --> 00:00:45.310
Well, 7 people are standing up,
no one is sitting in any of the

00:00:45.310 --> 00:00:49.480
chairs, so 7 different people
could sit in chair one.

00:00:49.480 --> 00:00:54.070
So there are 7 possibilities
for chair one.

00:00:54.070 --> 00:00:56.880
So 1 of those 7 possibilities
is going to sit

00:00:56.880 --> 00:00:58.400
down in chair one.

00:00:58.400 --> 00:01:02.060
So how many people are
left to sit in chair two?

00:01:02.060 --> 00:01:03.960
Well, 1 less than 7, right?

00:01:03.960 --> 00:01:07.310
6 people are going to be
left to sit in chair two.

00:01:07.310 --> 00:01:10.130
And then of course, if 2 people
are already sitting down, how

00:01:10.130 --> 00:01:13.480
many people are left to
sit in chair three?

00:01:13.480 --> 00:01:16.690
Well, there's going to
be 5 possibilities.

00:01:16.690 --> 00:01:19.640
So a way to think about it, for
each of these possibilities,

00:01:19.640 --> 00:01:23.940
there are 6 possibilities in
this chair and 5 possibilities

00:01:23.940 --> 00:01:25.980
in that chair for
each of those 6.

00:01:25.980 --> 00:01:27.760
So you would multiply all
of the possibilities.

00:01:27.760 --> 00:01:28.830
Hopefully that
makes some sense.

00:01:28.830 --> 00:01:31.320
We'll try it out
with some examples.

00:01:31.320 --> 00:01:33.140
So the total number of
possibilities in this

00:01:33.140 --> 00:01:37.170
case is 7 times 6 times
5, and what is that?

00:01:37.170 --> 00:01:39.500
6 times 5 is 30.

00:01:39.500 --> 00:01:42.950
30 times 7 equals 210.

00:01:42.950 --> 00:01:45.700
There's 210 possibilities.

00:01:45.700 --> 00:01:47.470
That's a lot of possibilities.

00:01:47.470 --> 00:01:50.000
Let's do a slightly
smaller example.

00:01:50.000 --> 00:01:51.530
I mean we don't have to
do people in chairs.

00:01:51.530 --> 00:01:52.770
Let's say we have cups.

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And I'll do the cups
in a different color.

00:01:58.880 --> 00:02:02.560
So this is 1 cup, cup one.

00:02:02.560 --> 00:02:05.300
And then I have cup two.

00:02:05.300 --> 00:02:06.720
And then I have 3 balls.

00:02:06.720 --> 00:02:12.610
I have a magenta ball, I
have a brown ball, and

00:02:12.610 --> 00:02:15.180
I have a yellow ball.

00:02:15.180 --> 00:02:18.570
And I want to know, how many
different ways can I put these

00:02:18.570 --> 00:02:21.620
3 balls into these 2 cups?

00:02:21.620 --> 00:02:23.820
Well, assuming that I haven't
put any of the balls into any

00:02:23.820 --> 00:02:26.130
of the cups as yet, how many
balls-- well, let's just

00:02:26.130 --> 00:02:27.750
start with cup two.

00:02:27.750 --> 00:02:30.410
I just want to show you that
you don't have to start at the

00:02:30.410 --> 00:02:34.620
cup that happens to be labeled
number one sitting on the left.

00:02:34.620 --> 00:02:38.380
If we haven't put any balls
into any of the cups, how

00:02:38.380 --> 00:02:39.790
many can we put in cup two?

00:02:39.790 --> 00:02:43.280
Well, there's 3
possibilities, right?

00:02:43.280 --> 00:02:45.720
And for each of those
3 then, how many can

00:02:45.720 --> 00:02:46.750
we put into cup one?

00:02:46.750 --> 00:02:49.240
Well, we would've put 1 into
here, so there will be 2

00:02:49.240 --> 00:02:51.300
left to put into cup two.

00:02:51.300 --> 00:02:54.480
So it'd be 3 times 2
possibilities, which

00:02:54.480 --> 00:02:57.420
is equal to 6.

00:02:57.420 --> 00:03:00.560
And let's see if we
can draw them out.

00:03:00.560 --> 00:03:02.010
Let me number these cups
because it's going to be

00:03:02.010 --> 00:03:04.820
too difficult to keep
switching colors.

00:03:04.820 --> 00:03:10.690
So let's say this
is A, B, and C.

00:03:10.690 --> 00:03:13.440
I'm going to show you that
there are 6 ways of putting

00:03:13.440 --> 00:03:18.040
these 3 balls into
these 2 cups.

00:03:18.040 --> 00:03:24.460
So it could be A in cup
one, B in cup two.

00:03:24.460 --> 00:03:27.770
It could be B in cup
one, A in cup two.

00:03:27.770 --> 00:03:28.740
Right, you could
just switch them.

00:03:28.740 --> 00:03:30.740
So we actually care which
ball goes into what cup.

00:03:30.740 --> 00:03:33.670
Not just is a ball in a cup.

00:03:33.670 --> 00:03:37.870
It could be A with C.

00:03:37.870 --> 00:03:39.800
A in cup one, C in cup two.

00:03:39.800 --> 00:03:43.270
It could be C in cup
one, A in cup two.

00:03:43.270 --> 00:03:45.390
And then finally, you
could just have B and C.

00:03:45.390 --> 00:03:48.530
B in cup one, C in cup two.

00:03:48.530 --> 00:03:51.780
Or C in cup one, B in cup two.

00:03:51.780 --> 00:03:56.440
And notice, if we just cared
which of the balls were picked,

00:03:56.440 --> 00:03:59.890
but we didn't care whether they
were in cup one or cup two, we

00:03:59.890 --> 00:04:02.270
could get rid of this bottom
line and there would be 1/2

00:04:02.270 --> 00:04:05.180
as many ways to place them.

00:04:05.180 --> 00:04:08.480
But when we care where the
balls are-- we care whether

00:04:08.480 --> 00:04:10.340
they're in cup one or cup two,
or in this case we care if

00:04:10.340 --> 00:04:12.530
they're in seat one,
two, or three.

00:04:12.530 --> 00:04:14.170
The ways in which we're
ordering, we call

00:04:14.170 --> 00:04:16.730
these permutations.

00:04:16.730 --> 00:04:22.060
And up here, the way we would
write this permutation, we

00:04:22.060 --> 00:04:26.050
would say, well, how many ways
can we fit 7 people into 3

00:04:26.050 --> 00:04:28.970
chairs if we care about what
chair they're sitting in?

00:04:28.970 --> 00:04:35.900
So we could write that as 7
people-- and the p isn't

00:04:35.900 --> 00:04:39.650
for people, it's for
permutations-- into 3 chairs.

00:04:39.650 --> 00:04:42.560
And another notation for it,
you write a big P and you

00:04:42.560 --> 00:04:46.380
say, how many ways can I
put 7 things into 3 spaces?

00:04:46.380 --> 00:04:50.220
Or it could be written as,
how many ways could you put

00:04:50.220 --> 00:04:53.100
7 things into 3 spaces?

00:04:53.100 --> 00:04:55.060
And in this case, we
got the answer is 210.

00:04:55.060 --> 00:04:58.790
And actually, it
will always be 210.

00:04:58.790 --> 00:05:00.920
And I'll show you
why in a second.

00:05:00.920 --> 00:05:02.160
How would we write this?

00:05:02.160 --> 00:05:05.990
Well, this would be--
I'll do it in green.

00:05:05.990 --> 00:05:09.520
We had 3 things and we
put it into 2 spaces.

00:05:09.520 --> 00:05:13.690
So 3 things into 2 spaces.

00:05:13.690 --> 00:05:17.770
That's the same thing as 3
things into 2 spaces, which

00:05:17.770 --> 00:05:21.160
is equal to 3 things
into 2 spaces.

00:05:21.160 --> 00:05:22.850
I'm just showing you the
different notations, and we

00:05:22.850 --> 00:05:25.480
figured out that that was 6.

00:05:25.480 --> 00:05:29.230
Let's see if we can figure
out a general formula if

00:05:29.230 --> 00:05:32.700
we wanted to evaluate nPk.

00:05:35.980 --> 00:05:40.805
How many ways can we fit
n things into k spaces?

00:05:43.310 --> 00:05:47.740
So if we think about it,
let's just use some analogy.

00:05:47.740 --> 00:05:52.940
We'll have k spots, so it'll
be 1 spot, 2 spots, 3 spots,

00:05:52.940 --> 00:05:56.980
all the way to k spots.

00:05:56.980 --> 00:06:01.790
In the first spot there's n
possibilities, just like we

00:06:01.790 --> 00:06:04.670
did in the last two examples.

00:06:04.670 --> 00:06:06.580
There's n possibilities there.

00:06:06.580 --> 00:06:10.980
Then, since 1 person or 1 thing
will be placed in spot one,

00:06:10.980 --> 00:06:14.300
there's going to be n minus 1
possibilities for spot two.

00:06:14.300 --> 00:06:19.660
And similarly, n minus 2
possibilities for spot three.

00:06:19.660 --> 00:06:22.360
And then, we could follow that
pattern all the way, and how

00:06:22.360 --> 00:06:24.855
many possibilities are there
going to be for spot k?

00:06:24.855 --> 00:06:28.520
Well in everything it's n
minus 1 thing less than the

00:06:28.520 --> 00:06:29.690
spot number in our case.

00:06:29.690 --> 00:06:32.150
I mean I guess we could have
started with spot 0, but this

00:06:32.150 --> 00:06:34.550
is better because we actually
numbered all the way to k.

00:06:34.550 --> 00:06:42.270
So it will be n
minus k minus 1.

00:06:42.270 --> 00:06:43.400
And that might seem
complicated, but

00:06:43.400 --> 00:06:45.000
it makes sense.

00:06:45.000 --> 00:06:55.350
When we did 7 things into 3
spots we went 7 times 6 times--

00:06:55.350 --> 00:07:00.130
we literally put in the top
three of you could almost

00:07:00.130 --> 00:07:01.800
say from the factorial.

00:07:01.800 --> 00:07:04.140
7 factorial will be
7 times 6 times 5.

00:07:04.140 --> 00:07:05.330
I think this will help.

00:07:05.330 --> 00:07:10.750
If this was just 3 was so
7 times 6 times 5 times 4

00:07:10.750 --> 00:07:12.350
times 3 times 2 times 1.

00:07:12.350 --> 00:07:13.650
That's 7 factorial.

00:07:13.650 --> 00:07:17.660
But we only took the first
3 of the factorial.

00:07:17.660 --> 00:07:21.110
And similarly, 6-- well,
6 is also 3 factorial.

00:07:21.110 --> 00:07:24.410
But another way you could view
it is we only took the first 2

00:07:24.410 --> 00:07:28.130
of 3 factorial So what we're
saying here is we just take

00:07:28.130 --> 00:07:30.230
the first k of n factorial.

00:07:30.230 --> 00:07:33.010
So is there any easy
way to write that?

00:07:33.010 --> 00:07:40.480
To write the first
k of n factorial.

00:07:40.480 --> 00:07:41.040
Why sure.

00:07:41.040 --> 00:07:47.020
We could write it as n
factorial and the formula's

00:07:47.020 --> 00:07:49.170
almost harder than
what it's saying.

00:07:49.170 --> 00:07:51.370
We could write it as n
factorial-- it;s a lower

00:07:51.370 --> 00:07:58.920
case n-- n factorial over
n minus k factorial.

00:07:58.920 --> 00:08:00.930
Let's see if that works.

00:08:00.930 --> 00:08:03.000
Well, what's going to happen?

00:08:03.000 --> 00:08:07.010
So in this case, n
factorial was 7.

00:08:07.010 --> 00:08:10.760
So it was 7 times 6
times 5 times 4 times

00:08:10.760 --> 00:08:14.020
3 times 2 times 1.

00:08:14.020 --> 00:08:16.890
And then what was n
minus k factorial?

00:08:16.890 --> 00:08:19.480
Well, in our example, k
was the number of spots,

00:08:19.480 --> 00:08:20.650
and there were 3 spots.

00:08:20.650 --> 00:08:25.440
So 7 minus 3 is 4,
so 4 factorial.

00:08:25.440 --> 00:08:28.590
4 times 3 times 2 times 1.

00:08:28.590 --> 00:08:30.460
And of course,
this will cancel.

00:08:30.460 --> 00:08:32.080
Let me do that in a
different color.

00:08:32.080 --> 00:08:36.940
This will cancel with this,
and so you're just left with

00:08:36.940 --> 00:08:41.090
the first 3 of 7 factorial.

00:08:41.090 --> 00:08:46.590
The 3 largest components
of 7 factorial.

00:08:46.590 --> 00:08:49.740
I always rederive this because
all I think of is what

00:08:49.740 --> 00:08:50.460
we did originally.

00:08:50.460 --> 00:08:51.500
I was like, oh, OK.

00:08:51.500 --> 00:08:55.530
If we're doing permutations,
I just start the factorial

00:08:55.530 --> 00:08:56.760
until I run out of spots.

00:08:56.760 --> 00:08:58.310
I do 7 times 6 times 5.

00:08:58.310 --> 00:09:01.290
And let's see if it
works out here too.

00:09:01.290 --> 00:09:05.450
So what is 3 factorial because
we have 3 different things?

00:09:05.450 --> 00:09:10.820
3 factorial over-- and
then 3 minus 2 factorial.

00:09:15.660 --> 00:09:18.190
Well, that's just 1 factorial.

00:09:18.190 --> 00:09:22.300
So it becomes 3 times
2 times 1 over 1.

00:09:22.300 --> 00:09:25.680
And of course, the 1's cancel
out and we got the 3 times 2.

00:09:25.680 --> 00:09:26.890
Hopefully that make sense.

00:09:26.890 --> 00:09:31.050
A lot of people, they learn the
permutation notation and they

00:09:31.050 --> 00:09:34.370
say, oh, that's n factorial
over n minus k factorial, where

00:09:34.370 --> 00:09:37.450
n is the number of things I
need to place and n minus

00:09:37.450 --> 00:09:38.630
the whole thing factorial.

00:09:38.630 --> 00:09:42.200
And k is the number
of spots I have.

00:09:42.200 --> 00:09:43.210
And they just memorize it.

00:09:43.210 --> 00:09:49.330
But all this is saying is you
start doing the n factorial,

00:09:49.330 --> 00:09:53.320
but you only do the first 3
components of the factorial.

00:09:53.320 --> 00:09:55.810
And it makes sense just from
how we started off the video.

00:09:55.810 --> 00:09:58.950
If you 7 things, well, 7 things
could go here, 6 things could

00:09:58.950 --> 00:10:00.740
go here, 5 things
could go here.

00:10:00.740 --> 00:10:01.580
No more spots?

00:10:01.580 --> 00:10:02.710
Just multiply them.

00:10:02.710 --> 00:10:05.140
And that's all this
formula is saying.

00:10:05.140 --> 00:10:07.400
Hopefully I didn't confuse you,
and I will see you in the

00:10:07.400 --> 00:10:09.590
next video on combinations.


Proof: d/dx(sqrt(x))

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:04.090
So I've been requested to do
the proof of the derivative of

00:00:04.090 --> 00:00:06.300
the square root of x, so I
thought I would do a quick

00:00:06.300 --> 00:00:08.300
video on the proof of the
derivative of the

00:00:08.300 --> 00:00:10.370
square root of x.

00:00:10.370 --> 00:00:13.680
So we know from the definition
of a derivative that the

00:00:13.680 --> 00:00:22.280
derivative of the function
square root of x, that is equal

00:00:22.280 --> 00:00:26.520
to-- let me switch colors, just
for a variety-- that's equal to

00:00:26.520 --> 00:00:33.080
the limit as delta
x approaches 0.

00:00:33.080 --> 00:00:35.595
And you know, some people
say h approaches 0,

00:00:35.595 --> 00:00:36.360
or d approaches 0.

00:00:36.360 --> 00:00:37.450
I just use delta x.

00:00:37.450 --> 00:00:39.450
So the change in x over 0.

00:00:39.450 --> 00:00:41.830
And then we say f of x
plus delta x, so in this

00:00:41.830 --> 00:00:42.910
case this is f of x.

00:00:42.910 --> 00:00:52.260
So it's the square root of x
plus delta x minus f of x,

00:00:52.260 --> 00:00:54.640
which in this case it's
square root of x.

00:00:54.640 --> 00:00:57.140
All of that over the change
in x, over delta x.

00:01:00.040 --> 00:01:02.580
Right now when I look at that,
there's not much simplification

00:01:02.580 --> 00:01:04.945
I can do to make this come out
with something meaningful.

00:01:09.940 --> 00:01:12.540
I'm going to multiply the
numerator and the denominator

00:01:12.540 --> 00:01:13.790
by the conjugate of the
numerator is what

00:01:13.790 --> 00:01:14.200
I mean by that.

00:01:14.200 --> 00:01:15.480
Let me rewrite it.

00:01:15.480 --> 00:01:19.740
Limit is delta x approaching
0-- I'm just rewriting

00:01:19.740 --> 00:01:21.280
what I have here.

00:01:21.280 --> 00:01:26.650
So I said the square root
of x plus delta x minus

00:01:26.650 --> 00:01:28.610
square root of x.

00:01:28.610 --> 00:01:31.200
All of that over delta x.

00:01:31.200 --> 00:01:34.490
And I'm going to multiply
that-- after switching colors--

00:01:34.490 --> 00:01:41.840
times square root of x plus
delta x plus the square root of

00:01:41.840 --> 00:01:48.260
x, over the square root of x
plus delta x plus the

00:01:48.260 --> 00:01:49.250
square root of x.

00:01:49.250 --> 00:01:53.420
This is just 1, so I could of
course multiply that times-- if

00:01:53.420 --> 00:01:57.110
we assume that x and delta x
aren't both 0, this is a

00:01:57.110 --> 00:01:59.090
defined number and
this will be 1.

00:01:59.090 --> 00:02:00.010
And we can do that.

00:02:00.010 --> 00:02:02.130
This is 1/1, we're just
multiplying it times this

00:02:02.130 --> 00:02:10.900
equation, and we get limit
as delta x approaches 0.

00:02:10.900 --> 00:02:13.510
This is a minus b
times a plus b.

00:02:13.510 --> 00:02:15.360
Let me do little aside here.

00:02:15.360 --> 00:02:20.880
Let me say a plus b times
a minus b is equal to a

00:02:20.880 --> 00:02:23.150
squared minus b squared.

00:02:23.150 --> 00:02:26.600
So this is a plus b
times a minus b.

00:02:26.600 --> 00:02:29.410
So it's going to be
equal to a squared.

00:02:29.410 --> 00:02:32.010
So what's this quantity squared
or this quantity squared,

00:02:32.010 --> 00:02:33.180
either one, these are my a's.

00:02:33.180 --> 00:02:35.450
Well it's just going
to be x plus delta x.

00:02:35.450 --> 00:02:39.430
So we get x plus delta x.

00:02:39.430 --> 00:02:41.050
And then what's b squared?

00:02:41.050 --> 00:02:46.380
So minus square root of
x is b in this analogy.

00:02:46.380 --> 00:02:50.640
So square root of x
squared is just x.

00:02:50.640 --> 00:02:56.760
And all of that over delta
x times square root of x

00:02:56.760 --> 00:03:04.210
plus delta x plus the
square root of x.

00:03:04.210 --> 00:03:05.900
Let's see what
simplification we can do.

00:03:05.900 --> 00:03:08.580
Well we have an x and
then a minus x, so those

00:03:08.580 --> 00:03:11.480
cancel out. x minus x.

00:03:11.480 --> 00:03:13.460
And then we're left in the
numerator and the denominator,

00:03:13.460 --> 00:03:15.690
all we have is a delta x here
and a delta x here, so let's

00:03:15.690 --> 00:03:18.770
divide the numerator and the
denominator by delta x.

00:03:18.770 --> 00:03:22.822
So this goes to 1,
this goes to 1.

00:03:22.822 --> 00:03:26.350
And so this equals the limit--
I'll write smaller, because I'm

00:03:26.350 --> 00:03:34.920
running out of space-- limit as
delta x approaches 0 of 1 over.

00:03:34.920 --> 00:03:37.780
And of course we can only do
this assuming that delta--

00:03:37.780 --> 00:03:40.220
well, we're dividing by delta
x to begin with, so we know

00:03:40.220 --> 00:03:42.420
it's not 0, it's just
approaching zero.

00:03:42.420 --> 00:03:50.320
So we get square root
of x plus delta x plus

00:03:50.320 --> 00:03:51.860
the square root of x.

00:03:51.860 --> 00:03:53.550
And now we can just
directly take the limit

00:03:53.550 --> 00:03:54.410
as it approaches 0.

00:03:54.410 --> 00:03:56.440
We can just set delta
x as equal to 0.

00:03:56.440 --> 00:03:58.140
That's what it's approaching.

00:03:58.140 --> 00:04:04.260
So then that equals one
over the square root of x.

00:04:04.260 --> 00:04:06.790
Right, delta x is 0, so
we can ignore that.

00:04:06.790 --> 00:04:09.120
We could take the limit
all the way to 0.

00:04:09.120 --> 00:04:13.000
And then this is of course just
a square root of x here plus

00:04:13.000 --> 00:04:17.160
the square root of x,
and that equals 1 over

00:04:17.160 --> 00:04:19.350
2 square root of x.

00:04:19.350 --> 00:04:24.890
And that equals 1/2x
to the negative 1/2.

00:04:24.890 --> 00:04:28.900
So we just proved that x to the
1/2 power, the derivative of it

00:04:28.900 --> 00:04:35.220
is 1/2x to the negative 1/2,
and so it is consistent with

00:04:35.220 --> 00:04:41.700
the general property that the
derivative of-- oh I don't

00:04:41.700 --> 00:04:50.850
know-- the derivative of x to
the n is equal to nx to the n

00:04:50.850 --> 00:04:55.150
minus 1, even in this case
where the n was 1/2.

00:04:55.150 --> 00:04:56.100
Well hopefully
that's satisfying.

00:04:56.100 --> 00:04:58.960
I didn't prove it for all
fractions but this is a start.

00:04:58.960 --> 00:05:01.120
This is a common one you
see, square root of x, and

00:05:01.120 --> 00:05:03.770
it's hopefully not too
complicated for proof.

00:05:03.770 --> 00:05:05.180
I will see you in
future videos.


Probability (part 8)

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.440
So in the last video I
hopefully proved to you, at

00:00:03.440 --> 00:00:06.720
least maybe it doesn't make
intuitive sense just yet, but I

00:00:06.720 --> 00:00:09.130
showed you that the probability
of a and b is equal to the

00:00:09.130 --> 00:00:11.400
probability of a given b
times the probability of b.

00:00:11.400 --> 00:00:13.700
And the probability of b given
a is equal to the probability

00:00:13.700 --> 00:00:15.250
of b given a times a.

00:00:15.250 --> 00:00:17.230
And then we played around a
little bit and we got this.

00:00:17.230 --> 00:00:19.550
And this is called Bayes'
theorem or Bayes' law.

00:00:19.550 --> 00:00:22.630
And I will now show you how
you can apply this formula.

00:00:22.630 --> 00:00:23.450
But it shouldn't
be in a formula.

00:00:23.450 --> 00:00:25.690
You should always be able to
intuitively come to this

00:00:25.690 --> 00:00:28.270
conclusion, even if you forget
Bayes' theorem, which I do all

00:00:28.270 --> 00:00:30.740
the time and then I go through
that thought process that

00:00:30.740 --> 00:00:33.050
we did at the end of the
last video over again.

00:00:33.050 --> 00:00:35.870
But let's see how we can
apply this to this problem.

00:00:35.870 --> 00:00:39.250
So we want to figure out the
probability that we pick the

00:00:39.250 --> 00:00:42.250
two-sided coin given that we
observed when we flip it

00:00:42.250 --> 00:00:44.670
five times we got 5 heads.

00:00:44.670 --> 00:00:51.920
So let's say that this is
event a and this is event b.

00:00:51.920 --> 00:00:53.720
So we can use Bayes' theorem.

00:00:53.720 --> 00:00:59.770
So the probability that I got
the two-sided coin, given 5 out

00:00:59.770 --> 00:01:07.440
of 5 heads is equal to the
probability-- and I'm actually

00:01:07.440 --> 00:01:09.930
going to think about what this
fraction means in a second.

00:01:09.930 --> 00:01:13.370
But it means the probability
that I got a given

00:01:13.370 --> 00:01:14.660
b, so it's this.

00:01:14.660 --> 00:01:19.760
The probability that I got 5
out of 5 heads, given the

00:01:19.760 --> 00:01:24.810
two-sided coin times the
probability of the two-sided

00:01:24.810 --> 00:01:28.150
coin-- b-- we defined event
b as the two-sided coin.

00:01:28.150 --> 00:01:30.830
Time the probability
of two-sided coin.

00:01:30.830 --> 00:01:35.870
Divided by the probability of
in general, the probability

00:01:35.870 --> 00:01:38.080
of getting 5 out of 5 heads.

00:01:43.610 --> 00:01:47.080
Well, what's the probability
of getting 5 out of 5 heads

00:01:47.080 --> 00:01:48.600
given the two-sided coin?

00:01:48.600 --> 00:01:49.680
Well, we wrote that down.

00:01:49.680 --> 00:01:54.580
We're guaranteed to get 5 out
of 5 heads, so that's 1.

00:01:54.580 --> 00:01:57.470
And what's the probability of
getting the two-sided coin?

00:01:57.470 --> 00:02:01.840
Well, that was 1/10.

00:02:01.840 --> 00:02:04.140
And now what's the probability
of getting 5 out of

00:02:04.140 --> 00:02:05.110
5 heads in general.

00:02:05.110 --> 00:02:08.660
Well, we figure that out, I
think, either in the last video

00:02:08.660 --> 00:02:10.720
or the video before that.

00:02:10.720 --> 00:02:12.170
I forgot the exact number.

00:02:12.170 --> 00:02:17.800
Actually, it was something
like 1/10 plus 9/320.

00:02:17.800 --> 00:02:20.910
Right, so this is 32.

00:02:20.910 --> 00:02:22.450
It was 41/320.

00:02:22.450 --> 00:02:25.760
We figured that out in
the previous problem.

00:02:25.760 --> 00:02:30.200
This is over 41 over 320.

00:02:30.200 --> 00:02:31.150
And so what did we get?

00:02:31.150 --> 00:02:38.190
We get this is equal
to 1/10 over 41/320.

00:02:38.190 --> 00:02:39.360
Now what is that equal to?

00:02:39.360 --> 00:02:45.670
That equals to 1/10
times 320/41.

00:02:45.670 --> 00:02:48.480
Let's divide the top
and the bottom by 10.

00:02:48.480 --> 00:02:50.510
So it equals 32/41.

00:02:50.510 --> 00:02:51.610
That's interesting.

00:02:51.610 --> 00:02:53.560
I actually find that
really interesting.

00:02:53.560 --> 00:02:56.150
I don't know which coin I
picked out of the bag, but I

00:02:56.150 --> 00:03:00.150
know that if I flip that coin
five times and all 5 are heads,

00:03:00.150 --> 00:03:04.360
there's a 32 out of 41 chance--
this is maybe a little bit

00:03:04.360 --> 00:03:10.880
better than a 75% chance that I
picked the two-headed

00:03:10.880 --> 00:03:12.330
coin we could say.

00:03:12.330 --> 00:03:15.490
And let's think about this
visually a little bit.

00:03:15.490 --> 00:03:20.120
So if this is the entire
universe of what could happen,

00:03:20.120 --> 00:03:25.830
and we said, 1/10 of that--
this is the two-sided coin,

00:03:25.830 --> 00:03:27.720
this is the fair coin.

00:03:27.720 --> 00:03:32.530
And then, out of all of that,
the situation in which we get 5

00:03:32.530 --> 00:03:37.350
out of 5, well, those could
be-- if we got the two-sided

00:03:37.350 --> 00:03:39.820
coin all of that is
going to be 5 out of 5.

00:03:39.820 --> 00:03:44.120
And then some small fraction,
1/32 of the fare coin.

00:03:44.120 --> 00:03:49.070
So this yellow circle
represents getting

00:03:49.070 --> 00:03:51.310
5 heads in a row.

00:03:51.310 --> 00:03:53.860
So if I want to know what's the
probability of the two-sided

00:03:53.860 --> 00:03:58.420
coin, all Bayes' theorem is
saying, well, OK, we observed

00:03:58.420 --> 00:04:00.490
that we got the two-sided coin.

00:04:00.490 --> 00:04:03.540
Sorry, we observed that
we got 5 out of 5 heads.

00:04:03.540 --> 00:04:06.980
So we now know that we're in
this yellow universe because

00:04:06.980 --> 00:04:11.470
the yellow universe represents
the area or all of the outcomes

00:04:11.470 --> 00:04:13.180
where we got 5 out of 5 heads.

00:04:13.180 --> 00:04:17.040
And then we say, well, out of
those outcomes, what percentage

00:04:17.040 --> 00:04:19.670
of them involved me getting
the two-sided coin?

00:04:19.670 --> 00:04:23.010
Well, if we look here,
all of these-- sorry.

00:04:23.010 --> 00:04:25.560
Edit.

00:04:25.560 --> 00:04:31.640
If we look at it here, this is
the area out of the yellow

00:04:31.640 --> 00:04:34.770
where I had the two-sided coin
and I got the 5 out of 5.

00:04:34.770 --> 00:04:40.300
And so this is going to
be roughly 32 out of 41

00:04:40.300 --> 00:04:41.320
of this yellow area.

00:04:41.320 --> 00:04:42.840
So that's a visual
representation of

00:04:42.840 --> 00:04:44.000
Bayes' theorem.

00:04:44.000 --> 00:04:45.810
This says the probability of
getting a two-sided coin,

00:04:45.810 --> 00:04:48.560
given I got 5 out of 5,
that says-- well, OK.

00:04:48.560 --> 00:04:51.660
Let's look at the universe of
you got 5 out of 5, and then,

00:04:51.660 --> 00:04:55.950
what percentage of that
universe involved you getting,

00:04:55.950 --> 00:04:57.370
essentially the two-sided coin?

00:04:57.370 --> 00:04:59.250
And that's that.

00:04:59.250 --> 00:05:04.710
And that's all Bayes' theorem
essentially tells us.

00:05:04.710 --> 00:05:07.260
So let's see if we
can apply that.

00:05:10.770 --> 00:05:13.690
Let's do a more
interesting one.

00:05:13.690 --> 00:05:16.090
Actually, I'm going to to leave
it there and then in the

00:05:16.090 --> 00:05:19.120
next video I will apply
Bayes' theorem to more

00:05:19.120 --> 00:05:20.025
interesting problems.

00:05:20.025 --> 00:05:21.740
See you soon.


Probability (part 7)

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.420
So let's think about what we
did in that last video and

00:00:03.420 --> 00:00:06.160
maybe do it in a little
bit more of a visual way.

00:00:06.160 --> 00:00:10.280
So let's say this rectangle
that I'm about to draw is

00:00:10.280 --> 00:00:19.660
the set of all the things
that I guess can happen.

00:00:19.660 --> 00:00:22.430
It's a set of all
of the outcomes.

00:00:22.430 --> 00:00:24.990
And we know from the get-go,
and we're still doing that same

00:00:24.990 --> 00:00:27.300
example where I have a bag of
coins and one of them

00:00:27.300 --> 00:00:28.890
only has heads on it.

00:00:28.890 --> 00:00:30.880
Two sides are heads, the
other ones are normal.

00:00:30.880 --> 00:00:33.090
So, if this is the whole set of
things we know that there's a

00:00:33.090 --> 00:00:37.060
1/10 chance that we pick the
coin that only has heads.

00:00:37.060 --> 00:00:38.510
So let me say out of
this area-- I don't

00:00:38.510 --> 00:00:40.370
know-- this is 1/2.

00:00:40.370 --> 00:00:42.690
That's 1/4.

00:00:42.690 --> 00:00:46.960
So 1/10 is going to be
about that much, roughly.

00:00:46.960 --> 00:00:48.280
So this area right here.

00:00:50.880 --> 00:00:52.520
And let me color it.

00:00:52.520 --> 00:00:54.690
Let me color it a color.

00:00:54.690 --> 00:01:00.410
So that area right there, this
is the chances that we got

00:01:00.410 --> 00:01:01.450
the two-sided sided coin.

00:01:04.620 --> 00:01:07.080
And this whole area, this
is all a normal coin.

00:01:10.090 --> 00:01:12.800
And then we said, oh, you know,
what are the chances that

00:01:12.800 --> 00:01:15.760
we get 5 heads in a row?

00:01:15.760 --> 00:01:19.350
Well, if we had a normal coin,
there was a 1 out of 32 chance

00:01:19.350 --> 00:01:20.880
that we get 5 heads in a row.

00:01:20.880 --> 00:01:24.670
So whatever this area
is, 1 out of 32.

00:01:24.670 --> 00:01:26.880
So I'll do it in a column.

00:01:26.880 --> 00:01:28.200
It's roughly there.

00:01:28.200 --> 00:01:30.460
It can be a little messy.

00:01:30.460 --> 00:01:36.790
And I'm assuming that this
column right here is 1/32

00:01:36.790 --> 00:01:38.060
of this normal area.

00:01:40.790 --> 00:01:46.370
So this is a situation of
normal-- I'll just write this

00:01:46.370 --> 00:01:51.090
as n-- and that upside down
U means and in set theory.

00:01:51.090 --> 00:01:52.380
Or it's a kind of intersection.

00:01:52.380 --> 00:01:55.100
But it means and I got
5 out of 5 heads.

00:02:00.800 --> 00:02:04.470
And if we think about,
what is the area of this?

00:02:04.470 --> 00:02:07.480
Well, it's going to be
the area-- it's 1/32

00:02:07.480 --> 00:02:09.550
of the normal area.

00:02:09.550 --> 00:02:12.570
And the normal area is 9/10.

00:02:12.570 --> 00:02:17.820
So this is equal to the
normal area, which is 9/10.

00:02:17.820 --> 00:02:20.400
And then 1/32 of that.

00:02:20.400 --> 00:02:24.710
Times 1/32.

00:02:24.710 --> 00:02:29.210
Or you could also, another way
of viewing that area, you could

00:02:29.210 --> 00:02:32.930
say that well, that area,
that's the probability-- I'll

00:02:32.930 --> 00:02:34.180
do it in the same color.

00:02:34.180 --> 00:02:39.240
That's the probability of
me getting-- I'll just

00:02:39.240 --> 00:02:40.140
write 5 out of 5.

00:02:40.140 --> 00:02:42.120
5 out of 5 means 5
out of 5 heads.

00:02:42.120 --> 00:02:44.590
That's what probability of
getting 5 out of 5 heads

00:02:44.590 --> 00:02:49.880
given that I picked
a normal coin out.

00:02:53.470 --> 00:03:00.450
Times the probability that I
picked out a normal coin.

00:03:00.450 --> 00:03:06.770
And so this is 1 out of 32,
and this is 9 out of 10.

00:03:06.770 --> 00:03:09.130
And I'm assuming that the
area of this entire magenta

00:03:09.130 --> 00:03:11.750
rectangle I already
drew, that's 1 or 100%.

00:03:11.750 --> 00:03:13.000
Or however you want to view it.

00:03:13.000 --> 00:03:15.150
And so this is 1 out of
32 times 9 out of 10.

00:03:15.150 --> 00:03:17.050
So this is 9 out of 32.

00:03:17.050 --> 00:03:19.430
So this whole thing, the area
this entire of this entire

00:03:19.430 --> 00:03:22.310
magenta rectangle, is 1.

00:03:22.310 --> 00:03:35.390
Then the area of this that I
just shaded in, this is 9/320.

00:03:35.390 --> 00:03:39.760
Another way of viewing this,
this is also the probability--

00:03:39.760 --> 00:03:43.920
this area right here is also
the probability that I got

00:03:43.920 --> 00:03:51.560
5 out of 5 and that I
picked the normal coin.

00:03:51.560 --> 00:03:53.820
So that's also equal to.

00:03:53.820 --> 00:03:56.430
Both of those things
are true in this area.

00:03:56.430 --> 00:03:59.560
In this area, if I have a point
in that area-- this might be a

00:03:59.560 --> 00:04:02.210
little abstract for you-- if at
any point, it is true that I

00:04:02.210 --> 00:04:06.100
got 5 out of 5, and I have
picked a normal coin.

00:04:06.100 --> 00:04:08.630
So that's also going to be
equal to and I'm doing an

00:04:08.630 --> 00:04:10.510
arbitrary color change here.

00:04:10.510 --> 00:04:13.310
9 out of 320.

00:04:13.310 --> 00:04:19.390
So we could say in general,
that the probability of me

00:04:19.390 --> 00:04:25.920
getting 5 out of 5 coins, given
that I got a normal coin times

00:04:25.920 --> 00:04:28.960
the probability that I got the
normal coin-- that is going

00:04:28.960 --> 00:04:34.160
to be equal that 5 out of 5
happens and I got the normal.

00:04:34.160 --> 00:04:39.920
That equals probability
of 5 out of 5, and I

00:04:39.920 --> 00:04:41.600
got the normal coin.

00:04:41.600 --> 00:04:43.810
Fair enough.

00:04:43.810 --> 00:04:45.350
Let's go to the
other side of this.

00:04:45.350 --> 00:04:47.670
This kind of yellowish
color here.

00:04:47.670 --> 00:04:49.610
There's a probability I
got a two-sided coin.

00:04:49.610 --> 00:04:52.125
And then given that, what's
the probability that

00:04:52.125 --> 00:04:53.630
I get 5 out of 5?

00:04:53.630 --> 00:04:55.800
Well, pretty much,
this entire area.

00:04:55.800 --> 00:04:58.930
It's also true that
I get 5 out of 5.

00:05:02.060 --> 00:05:14.190
So the probability that I get 5
out of 5 and two-sided, it's

00:05:14.190 --> 00:05:17.640
equal to the probability--
oh sorry, and.

00:05:17.640 --> 00:05:22.600
And two-sided, is equal to the
probability of 5 out of 5,

00:05:22.600 --> 00:05:26.290
given the two-sided coin
times the probability

00:05:26.290 --> 00:05:28.710
of the two-sided coin.

00:05:28.710 --> 00:05:31.200
So this is 1, right?

00:05:31.200 --> 00:05:33.610
If I got the two-sided coin I'm
definitely going to 5 out of 5,

00:05:33.610 --> 00:05:36.670
That's why this whole
area is filled in.

00:05:36.670 --> 00:05:38.410
And then, the probability of
the two-sided coin we know

00:05:38.410 --> 00:05:40.410
already is 1 out of 10.

00:05:40.410 --> 00:05:42.090
So this area is 1 out of 10.

00:05:44.980 --> 00:05:48.840
And so what is the total area
where I have 5 out of 5?

00:05:48.840 --> 00:05:51.340
Well, it's going to be
this area over here,

00:05:51.340 --> 00:05:53.070
which I figured out.

00:05:53.070 --> 00:05:58.530
Which was 9/320 of this
entire rectangle.

00:05:58.530 --> 00:06:02.250
And then this whole area is
also another area where I

00:06:02.250 --> 00:06:05.460
got 5 out of 5 because I
picked the two-sided coin.

00:06:05.460 --> 00:06:07.640
And that's 1/10.

00:06:07.640 --> 00:06:10.020
In the last video we would
essentially-- if we want to

00:06:10.020 --> 00:06:12.770
know the chances of getting
5 out of 5, it's this

00:06:12.770 --> 00:06:14.960
total rectangle.

00:06:14.960 --> 00:06:16.937
I say this without showing
you anything, so let

00:06:16.937 --> 00:06:18.120
me-- it'll be red.

00:06:21.840 --> 00:06:23.280
And I think you get the point.

00:06:23.280 --> 00:06:25.000
That is the 5 out of 5,
so it's going to be this

00:06:25.000 --> 00:06:26.600
area plus this area.

00:06:26.600 --> 00:06:34.340
And that is why it was 9 out
of 320 plus 1 out of 10.

00:06:34.340 --> 00:06:35.890
And we had the
answer last time.

00:06:35.890 --> 00:06:37.130
Fair enough.

00:06:37.130 --> 00:06:43.350
So let me pose a slightly
more difficult problem.

00:06:43.350 --> 00:06:45.850
I don't know if I should have
just erased that, but I did.

00:06:45.850 --> 00:06:47.280
So let's move forward.

00:06:47.280 --> 00:06:52.780
What is the probability that I
picked the two-sided coin given

00:06:52.780 --> 00:06:54.530
that I got 5 out of 5 heads?

00:06:57.170 --> 00:07:00.370
And remember, this is different
than the probability that I get

00:07:00.370 --> 00:07:04.270
5 out of 5 heads given
the two-sided coin.

00:07:04.270 --> 00:07:05.230
What's the difference?

00:07:05.230 --> 00:07:07.680
This says I picked the
two-sided coin, now what's the

00:07:07.680 --> 00:07:09.340
probability I got 5 out of 5?

00:07:09.340 --> 00:07:10.680
Well that's equal to 1.

00:07:10.680 --> 00:07:13.230
We're guaranteed to get a
million heads in a row.

00:07:13.230 --> 00:07:17.130
But this says I put my hand in
the bag, I didn't look at which

00:07:17.130 --> 00:07:20.660
coin I got, I just flipped it
five times and I know that

00:07:20.660 --> 00:07:22.860
I got 5 heads in a row.

00:07:22.860 --> 00:07:25.720
So now I say, well, what is the
probability without looking at

00:07:25.720 --> 00:07:28.070
the coin, what is the
probability that this was

00:07:28.070 --> 00:07:30.830
the two-sided coin
that I picked up?

00:07:30.830 --> 00:07:33.590
Well, we can go back to
what we said before.

00:07:33.590 --> 00:07:36.150
Let me just say it a
little bit abstract.

00:07:36.150 --> 00:07:41.920
Before I clear this, we know
that the probability of a

00:07:41.920 --> 00:07:49.860
given b, or the probability
of-- well, actually.

00:07:49.860 --> 00:07:51.000
Let me start over.

00:07:51.000 --> 00:07:55.300
The probability of a and b
happening, if you remember that

00:07:55.300 --> 00:07:59.270
thing that before I erased it,
is a probability of a given

00:07:59.270 --> 00:08:04.050
that b happened times the
probability that b happens.

00:08:04.050 --> 00:08:07.820
If both of these things have to
happen then we could say, well,

00:08:07.820 --> 00:08:10.300
definitely b has to happen.

00:08:10.300 --> 00:08:12.780
And then, whatever the
probability of a happening

00:08:12.780 --> 00:08:13.690
given that b happened.

00:08:13.690 --> 00:08:17.060
This could be exclusive events.

00:08:17.060 --> 00:08:19.790
If these are independent
this could be equal to the

00:08:19.790 --> 00:08:22.620
probability of a times
the probability of b.

00:08:22.620 --> 00:08:25.390
But we don't know for sure that
they're exclusive events.

00:08:25.390 --> 00:08:28.210
So the probability of a given b
could be the probability of a.

00:08:28.210 --> 00:08:30.330
It might not be dependent on b
at all, but we don't know that.

00:08:30.330 --> 00:08:32.470
Let's ignore that.

00:08:32.470 --> 00:08:34.630
So the probability of
a given b is that.

00:08:34.630 --> 00:08:39.540
But we also know that the
probability of a given

00:08:39.540 --> 00:08:45.540
b is equal to-- I mean,
the probability-- sorry.

00:08:45.540 --> 00:08:48.350
We also know the probability of
a and b is also the same thing

00:08:48.350 --> 00:08:51.580
as the probability of
b and a happening.

00:08:51.580 --> 00:08:52.580
The same thing.

00:08:52.580 --> 00:08:56.630
And so that's the probability
of b given that a happened,

00:08:56.630 --> 00:09:00.150
times the probability
that a happened.

00:09:00.150 --> 00:09:01.670
By the same logic.

00:09:01.670 --> 00:09:04.900
Well, we see that this is
equal to this, so this

00:09:04.900 --> 00:09:06.500
has to be equal to that.

00:09:06.500 --> 00:09:08.212
So let's write that down.

00:09:08.212 --> 00:09:09.960
And we're going to do a lot of
applications; I know this is

00:09:09.960 --> 00:09:12.440
a little abstract right now.

00:09:12.440 --> 00:09:15.920
So that tells us the
probability of a given b times

00:09:15.920 --> 00:09:22.990
the probability of b, is equal
to the probability of b given

00:09:22.990 --> 00:09:26.690
a, times the probability of a.

00:09:26.690 --> 00:09:30.810
And then, let's just
divide both sides by a.

00:09:30.810 --> 00:09:36.160
And we get this, the
probability of b given

00:09:36.160 --> 00:09:43.490
a is equal to this,
probability of a given b.

00:09:43.490 --> 00:09:47.390
Times the probability
of b divided by the

00:09:47.390 --> 00:09:49.120
probability of a.

00:09:49.120 --> 00:09:51.600
And this is called Bayes'
theorem or Bayes' law.

00:09:51.600 --> 00:09:55.210
And this is what we will now
use to figure out the

00:09:55.210 --> 00:09:58.930
probability that we got the
two-sided coin after seeing

00:09:58.930 --> 00:10:02.930
that we rolled or we
got 5 heads in a row.

00:10:02.930 --> 00:10:04.870
See you in the next video.


Probability (part 6)

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.070
Now our probability problems
will get a little bit

00:00:03.070 --> 00:00:03.860
more interesting.

00:00:03.860 --> 00:00:06.700
So let's say I have a
bag full of coins.

00:00:06.700 --> 00:00:12.360
So this is my bag of coins,
maybe it's a bowl of coins.

00:00:12.360 --> 00:00:14.690
From a bunch of coins in there.

00:00:14.690 --> 00:00:16.820
I didn't need to draw
it, but I did anyway.

00:00:16.820 --> 00:00:21.510
And let's say that I know that
9 of these coins are normal.

00:00:24.100 --> 00:00:25.390
So what do I mean my normal?

00:00:25.390 --> 00:00:27.210
Just, you know, let's say
they're quarters and they have

00:00:27.210 --> 00:00:29.580
a heads and a tails, and that
there's a 50% chance of getting

00:00:29.580 --> 00:00:32.260
a heads or 50% chance of
a tails if you flip

00:00:32.260 --> 00:00:33.640
any of those coins.

00:00:33.640 --> 00:00:38.090
And then, 1 of the coins
is a two-sided head coin.

00:00:41.640 --> 00:00:44.110
So essentially, no matter
what you flip it, it's

00:00:44.110 --> 00:00:47.070
going to be a heads.

00:00:47.070 --> 00:00:49.740
So I have this bag, what I'm
going to do is I'm going to

00:00:49.740 --> 00:00:52.820
blindfold myself, stick my
hand in the bag, and

00:00:52.820 --> 00:00:54.380
pick a coin out.

00:00:54.380 --> 00:00:58.230
And then I'm going to flip
that coin, I'm going

00:00:58.230 --> 00:01:00.950
to flip it five times.

00:01:00.950 --> 00:01:05.590
And my question to you is,
what is the probability that

00:01:05.590 --> 00:01:08.515
I get all 5 to be heads?

00:01:19.170 --> 00:01:22.400
So whenever you see something
unfamiliar it'll always work--

00:01:22.400 --> 00:01:24.620
maybe we'll do a tree or we'll
do a hybrid of a probability

00:01:24.620 --> 00:01:25.500
tree or something else.

00:01:25.500 --> 00:01:28.630
But you say, well, at the very
beginning of this experiment

00:01:28.630 --> 00:01:29.720
something happens.

00:01:29.720 --> 00:01:32.880
I'm going to pick a
coin out of this bag.

00:01:32.880 --> 00:01:37.350
And so you could say that
there's a-- well, how

00:01:37.350 --> 00:01:38.050
many coins are there?

00:01:38.050 --> 00:01:40.630
There's 10 coins and I have an
equal probability of picking

00:01:40.630 --> 00:01:41.355
any one of the coins.

00:01:41.355 --> 00:01:43.690
Let's say I can't feel
whether it's heads or tails.

00:01:43.690 --> 00:01:47.250
I'm doing it with gloves
on, so I can't feel out

00:01:47.250 --> 00:01:49.280
the two-sided coin.

00:01:49.280 --> 00:01:51.150
Take my word for it, there's
an equal probability to take

00:01:51.150 --> 00:01:53.440
any 1 of these 10 coins.

00:01:53.440 --> 00:01:57.370
There's 9 normal, 1 weird
with 2 heads on it.

00:01:57.370 --> 00:01:59.320
So what's the probability
that I get a normal coin?

00:02:02.860 --> 00:02:06.530
Well, there are 9 normal
coins out of the 10 equally

00:02:06.530 --> 00:02:07.830
likely coins I could get.

00:02:07.830 --> 00:02:11.560
So the probability of a
normal is 9 out of 10.

00:02:11.560 --> 00:02:15.755
So there's a 9 out of 10 chance
that this is a normal coin

00:02:15.755 --> 00:02:18.890
Let's just call that normal.

00:02:18.890 --> 00:02:21.560
And then, what's the
probability that I get

00:02:21.560 --> 00:02:24.930
a-- let's call it a
2s-- two-sided coin.

00:02:24.930 --> 00:02:26.080
I'll call it 2s.

00:02:26.080 --> 00:02:27.800
Well, that's just 1 out of 10.

00:02:27.800 --> 00:02:29.670
I'm either going to
get this or that.

00:02:29.670 --> 00:02:35.060
I'm not going to take nothing
out of the bag I assume.

00:02:35.060 --> 00:02:40.630
OK, so this is a 1/10 chance
that I get a two-sided coin.

00:02:40.630 --> 00:02:43.280
And then whichever coin
I get I'm going to flip

00:02:43.280 --> 00:02:45.620
it five times in a row.

00:02:45.620 --> 00:02:47.010
And I'm not going to do
the tree here because

00:02:47.010 --> 00:02:50.720
to drew a tree I'd have
32 nodes on each side.

00:02:50.720 --> 00:02:54.450
But given this, given
the normal, what is

00:02:54.450 --> 00:02:58.220
now the probability of
getting 5 in a row?

00:02:58.220 --> 00:03:05.110
The way I'm going to write that
is the probability of 5 heads--

00:03:05.110 --> 00:03:07.690
or I could have written 5
out of 5 heads, whatever.

00:03:07.690 --> 00:03:11.120
And then this upper line, this
means given-- given that

00:03:11.120 --> 00:03:13.370
I have a normal coin.

00:03:13.370 --> 00:03:15.620
So if I can assume that I had a
normal coin, what are the

00:03:15.620 --> 00:03:18.180
probabilities that my next 5--
so I'll say this is

00:03:18.180 --> 00:03:20.330
5 out of 5 heads.

00:03:20.330 --> 00:03:21.380
Well, we figure that out.

00:03:21.380 --> 00:03:24.900
Each time there's a 1/2
probability and I'm going

00:03:24.900 --> 00:03:26.520
to do it 5 in a row.

00:03:26.520 --> 00:03:32.800
So it's 1/2 to the fifth,
which equals 1/32.

00:03:32.800 --> 00:03:35.440
So the probability that I get
5 out of 5 heads given a

00:03:35.440 --> 00:03:38.280
normal coin is 1 out of 32.

00:03:38.280 --> 00:03:39.570
Fair enough.

00:03:39.570 --> 00:03:47.520
Well, what's the probability
of 5 out of 5 heads,

00:03:47.520 --> 00:03:49.890
given the two-sided coin?

00:03:49.890 --> 00:03:51.750
And now just is this random
notation I'm coming

00:03:51.750 --> 00:03:52.370
with the two-sided.

00:03:52.370 --> 00:03:54.960
But this upwards and downwards
line, this vertical

00:03:54.960 --> 00:03:56.310
line, that means given.

00:04:01.360 --> 00:04:04.490
I mean I could flip this coin
a million times and all of

00:04:04.490 --> 00:04:05.830
them are going to be heads.

00:04:05.830 --> 00:04:09.630
So the probability
here is 1 out of 1.

00:04:09.630 --> 00:04:11.380
I'm definitely going
to get a heads here.

00:04:17.040 --> 00:04:20.770
So all I asked initially is,
what is the probability that

00:04:20.770 --> 00:04:24.190
I get 5 out of 5 heads?

00:04:24.190 --> 00:04:26.730
Well, this is going to be equal
to-- and I'll switch colors

00:04:26.730 --> 00:04:29.510
arbitrarily here just to
keep things interesting.

00:04:29.510 --> 00:04:36.230
This is going to be equal to
the probability that you get 5

00:04:36.230 --> 00:04:44.400
out of 5 heads given-- let's
call that a normal coin-- given

00:04:44.400 --> 00:04:47.170
a normal coin times the
probability that you had a

00:04:47.170 --> 00:04:50.110
normal coin times
the probability of

00:04:50.110 --> 00:04:50.740
let's say normal.

00:04:54.730 --> 00:04:58.090
Plus-- and I'm running a little
out of space-- the probability

00:04:58.090 --> 00:05:06.390
that you get 5 out of 5 heads
given the two-sided coin times

00:05:06.390 --> 00:05:10.660
the probability that you had
the two-sided sided coin.

00:05:10.660 --> 00:05:12.710
Hopefully that makes sense and
you kind of think of it in this

00:05:12.710 --> 00:05:16.140
way, what is the probability
of getting to this outcome?

00:05:16.140 --> 00:05:20.510
It is the probability that you
picked a normal coin and then,

00:05:20.510 --> 00:05:23.780
given the normal coin,
there's a 1 in 32 chance

00:05:23.780 --> 00:05:24.630
that you got there.

00:05:24.630 --> 00:05:26.670
So it'd be 1 out of 32
times the chance you've

00:05:26.670 --> 00:05:27.520
got the normal coin.

00:05:27.520 --> 00:05:29.340
So 9 out of 10.

00:05:29.340 --> 00:05:30.820
And what's the probability
of getting here?

00:05:30.820 --> 00:05:33.700
What's the probability you had
to-- and, in general, in a

00:05:33.700 --> 00:05:36.390
tree, you just have to multiply
all of the nodes to get to that

00:05:36.390 --> 00:05:38.400
point to figure out what's the
probability of getting there.

00:05:38.400 --> 00:05:41.000
And I think if you watched
the first few videos it make

00:05:41.000 --> 00:05:42.690
sense why that happens.

00:05:42.690 --> 00:05:44.090
So what's the
probability of this?

00:05:44.090 --> 00:05:46.490
Well, there's a 1/10 chance
that you picked the

00:05:46.490 --> 00:05:47.670
two-sided coin.

00:05:47.670 --> 00:05:51.410
And then of course, you're
guaranteed to get all heads.

00:05:51.410 --> 00:05:54.870
This is 1 times 1/10.

00:05:54.870 --> 00:05:56.550
And all I did here and
here is the same thing

00:05:56.550 --> 00:05:57.440
that I wrote up here.

00:06:00.850 --> 00:06:02.990
Let me switch colors.

00:06:02.990 --> 00:06:09.720
So what is this, the
probability of 5 out of 5 heads

00:06:09.720 --> 00:06:12.290
given a normal coin times the
probability of a normal coin?

00:06:12.290 --> 00:06:13.320
Well, that's just this.

00:06:13.320 --> 00:06:15.620
This is the probability of 5
out of 5 heads given the normal

00:06:15.620 --> 00:06:18.260
coin times the probability
of a normal coin.

00:06:18.260 --> 00:06:19.340
And this is equal to what?

00:06:19.340 --> 00:06:24.590
9/320.

00:06:24.590 --> 00:06:28.230
And then this, the probability
of 5 out of 5 heads given the

00:06:28.230 --> 00:06:29.810
two-sided coin times the
probability of the

00:06:29.810 --> 00:06:30.980
two-sided coin.

00:06:30.980 --> 00:06:33.970
Well, this is the probability
of 5 out of 5 heads given

00:06:33.970 --> 00:06:36.400
the two-sided coin, you're
guaranteed to get 5 out of 5.

00:06:36.400 --> 00:06:38.260
And the probability of
getting a two-sided

00:06:38.260 --> 00:06:40.560
coin is 1 out of 10.

00:06:40.560 --> 00:06:42.620
This is this.

00:06:42.620 --> 00:06:44.850
So this is equal to 1/10.

00:06:44.850 --> 00:06:49.070
So my probability of just
putting my hand in the bag,

00:06:49.070 --> 00:06:51.630
taking out a random coin and
then flipping it five times and

00:06:51.630 --> 00:06:57.270
getting all 5 heads, is going
to be equal to this plus this.

00:06:57.270 --> 00:07:03.100
So it's 9/320.

00:07:03.100 --> 00:07:05.180
plus 1/10.

00:07:05.180 --> 00:07:08.433
But let's make
that so it's 320.

00:07:08.433 --> 00:07:12.320
That's 32/320.

00:07:12.320 --> 00:07:17.790
So 9 plus 32 is 41/320.

00:07:17.790 --> 00:07:19.560
So that's interesting.

00:07:19.560 --> 00:07:24.390
But now let me ask you even a
more interesting question.

00:07:24.390 --> 00:07:29.390
This is why I think what we've
learned so far is useful.

00:07:29.390 --> 00:07:32.120
Let's say that I do
get 5 out of 5 heads.

00:07:32.120 --> 00:07:34.370
So I know that I got
5 out of 5 heads.

00:07:34.370 --> 00:07:36.910
So let me clear this up a
little bit just because I think

00:07:36.910 --> 00:07:41.540
it's getting over-- So let's
say that I do get

00:07:41.540 --> 00:07:42.980
5 out of 5 heads.

00:07:42.980 --> 00:07:47.140
What is the probability that
I picked the two-sided coin?

00:07:47.140 --> 00:07:52.570
So what we're asking is, what
is the probability that I

00:07:52.570 --> 00:07:57.080
picked the two-sided coin,
given that I got 5

00:07:57.080 --> 00:07:57.990
out of 5 heads?

00:08:01.280 --> 00:08:04.290
And I hope you understand
what I mean here.

00:08:04.290 --> 00:08:07.380
Before we said, oh, for
example, I hope you see the

00:08:07.380 --> 00:08:09.660
difference between this and
what is the probability of

00:08:09.660 --> 00:08:15.840
getting 5 out of 5 heads,
given the two-sided coin.

00:08:15.840 --> 00:08:18.210
If I know I have a two-sided
coin, I know that the

00:08:18.210 --> 00:08:21.500
probability is equal to 1.

00:08:21.500 --> 00:08:23.710
I know that I'm guaranteed
to get 5 heads.

00:08:23.710 --> 00:08:25.430
But this is a little
bit more interesting.

00:08:25.430 --> 00:08:27.650
If I flip a coin five times in
a row and I get a bunch of

00:08:27.650 --> 00:08:31.600
heads, I'll say, well, I either
got maybe really lucky with a

00:08:31.600 --> 00:08:35.350
normal coin or I picked
the two-sided coin.

00:08:35.350 --> 00:08:37.110
And I don't know which
one happened, so

00:08:37.110 --> 00:08:40.030
there's a probability.

00:08:40.030 --> 00:08:42.430
And of course, the more times
you get a heads, just

00:08:42.430 --> 00:08:44.820
intuitively, if I got 100 heads
in a row I'll say, well, I

00:08:44.820 --> 00:08:48.340
probably picked the two-sided
coin without actually just

00:08:48.340 --> 00:08:49.140
looking at the coin.

00:08:49.140 --> 00:08:51.000
We're assuming that
that's not acceptable.

00:08:51.000 --> 00:08:53.680
And if I got a million heads in
a row we'd be really, really

00:08:53.680 --> 00:08:55.370
sure that we picked the
two-sided coin, but we're

00:08:55.370 --> 00:08:56.380
not completely sure.

00:08:56.380 --> 00:08:58.940
There's some probability that I
picked the fair coin and it

00:08:58.940 --> 00:09:01.690
just happened to get heads
a million times in a row.

00:09:01.690 --> 00:09:04.170
So how do we think about this?

00:09:04.170 --> 00:09:08.500
Well, I'll do it analytically
and I'll draw it out, and

00:09:08.500 --> 00:09:11.400
hopefully, both of them
will make sense to you.

00:09:11.400 --> 00:09:13.950
And I'll probably have to
do it in the next video.

00:09:13.950 --> 00:09:15.672
I'll see you soon.


Probability (part 5)

WEBVTT
Kind: captions
Language: en

00:00:04.180 --> 00:00:08.100
let's say we're playing monopoly with two six sided dice and I say what is the probability that I will get

00:00:08.100 --> 00:00:15.650
a 7? when I add up the two rolls of the dice so I drew this grid here:and this grid esentially represents

00:00:15.650 --> 00:00:23.591
all the outcomes I can get with the two dice where on the top row that is the outcomeI can get on dice

00:00:23.591 --> 00:00:27.446
one: I can get a 1, a 2, a 3, a 4, a 5, or a 6 and similarlyfor dice two these are all of the outcomes


Probability (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.930
So we started doing some
free-throw percentage

00:00:02.930 --> 00:00:05.060
problems in the last
video, so let's continue.

00:00:05.060 --> 00:00:08.290
So the first time I said if
I had an 80% free-throw

00:00:08.290 --> 00:00:11.000
percentage, what are my
chances of getting three

00:00:11.000 --> 00:00:12.290
free throws in a row?

00:00:12.290 --> 00:00:15.640
And I said, well,
it's 51.2% chance.

00:00:15.640 --> 00:00:18.020
Let's do another scenario, and
let's just assume that my

00:00:18.020 --> 00:00:21.080
free-throw percentage
stays at 80%.

00:00:21.080 --> 00:00:29.050
And let's say that-- I don't
know-- let's say my team

00:00:29.050 --> 00:00:30.320
is two points behind.

00:00:30.320 --> 00:00:32.519
I'm making this up on the fly,
so we're two points behind.

00:00:36.960 --> 00:00:38.360
No, let's say we're
one point behind.

00:00:46.420 --> 00:00:50.310
And I'm up at the free-throw
line, and I want to say, what

00:00:50.310 --> 00:00:57.520
is the chance-- and let's say
I have three free throws.

00:00:57.520 --> 00:01:00.440
Someone fouled me when I
was shooting a 3-pointer.

00:01:00.440 --> 00:01:06.960
So we're one point behind,
there's one second left, and

00:01:06.960 --> 00:01:10.820
right when I was going to take
the winning free throw-- or

00:01:10.820 --> 00:01:11.680
not the winning free throw.

00:01:11.680 --> 00:01:14.500
The winning 3-point
shot, someone fouls me.

00:01:14.500 --> 00:01:16.245
So I get three free throws.

00:01:19.950 --> 00:01:23.110
And once again, I have an
80% free-throw percentage.

00:01:23.110 --> 00:01:24.690
My actual percentage
is probably something

00:01:24.690 --> 00:01:26.390
more like 20%.

00:01:26.390 --> 00:01:29.660
But anyway, I have
three free throws.

00:01:29.660 --> 00:01:33.980
So my question is, what are
our chances of winning?

00:01:33.980 --> 00:01:37.730
What is the probability
of my team winning?

00:01:40.850 --> 00:01:43.580
Not tying, winning.

00:01:43.580 --> 00:01:46.760
So in order to win--
well, actually let's

00:01:46.760 --> 00:01:48.950
do the tying first.

00:01:48.950 --> 00:01:50.320
So what's the
probability of tying?

00:01:52.870 --> 00:01:55.840
How do you spell-- tying, OK.

00:01:55.840 --> 00:01:59.190
What's the probability
of a tie?

00:01:59.190 --> 00:02:04.740
Well, the probability of a tie
is going to be equal to the

00:02:04.740 --> 00:02:17.600
probability of me getting
at least one free throw.

00:02:17.600 --> 00:02:21.870
You might say OK, that's the
probably of getting exactly 1

00:02:21.870 --> 00:02:24.940
times the probability of
getting 2 out of 3 plus

00:02:24.940 --> 00:02:27.360
the probability of
getting 3 out of 3.

00:02:27.360 --> 00:02:30.280
But like we saw in-- I think it
was either the last video or

00:02:30.280 --> 00:02:34.820
the video before-- that is
identical equal to the

00:02:34.820 --> 00:02:39.590
probability or, let's see, 1
minus the probability of not

00:02:39.590 --> 00:02:41.080
getting any free throws.

00:02:49.580 --> 00:02:52.950
The probability of getting at
least one is essentially the

00:02:52.950 --> 00:02:56.220
probability of me not
missing all three.

00:02:56.220 --> 00:02:57.600
Hopefully that
makes some sense.

00:02:57.600 --> 00:03:01.500
So let me erase all this stuff
that I did up on top, and

00:03:01.500 --> 00:03:04.440
we can to figure out the
answer to this problem.

00:03:04.440 --> 00:03:08.910
Let me erase all of
this, erase all that.

00:03:08.910 --> 00:03:10.630
Erase that.

00:03:10.630 --> 00:03:13.180
Let me erase this.

00:03:13.180 --> 00:03:16.160
We know I have an 80%
probability of making

00:03:16.160 --> 00:03:18.230
any one shot.

00:03:18.230 --> 00:03:22.930
So if we go back to the
problem-- so this is my

00:03:22.930 --> 00:03:25.740
free throw percentage.

00:03:25.740 --> 00:03:31.940
So what is the probability of
me not making any shots, not

00:03:31.940 --> 00:03:37.850
making any of my free throws?

00:03:37.850 --> 00:03:42.830
Well, my probability of not
making one free throw is 20%.

00:03:42.830 --> 00:03:47.920
20% chance of not making any
free throws, or not making

00:03:47.920 --> 00:03:54.210
any-- say I take a shot, I have
a 20% of no basket, or not

00:03:54.210 --> 00:03:57.290
making the free throw.

00:03:57.290 --> 00:04:01.310
So if I'm taking three shots,
the shot of I miss all three

00:04:01.310 --> 00:04:11.000
is going to be equal to
20% times 20% times 20%.

00:04:11.000 --> 00:04:12.420
And that equals what?

00:04:12.420 --> 00:04:19.550
This 0.2 times 0.2
is 0.04 times 0.2.

00:04:22.280 --> 00:04:23.530
And then let's see.

00:04:23.530 --> 00:04:25.960
What's 0.04 times 0.2?

00:04:25.960 --> 00:04:29.790
Let's see, that'll be an 8
and I have 1, 2, 3 numbers

00:04:29.790 --> 00:04:30.620
by the decimal place.

00:04:30.620 --> 00:04:32.870
It's 1, 2, 3.

00:04:32.870 --> 00:04:34.930
So 0.008.

00:04:34.930 --> 00:04:42.340
So I have a 0.008 chance of
missing all of the free throws.

00:04:42.340 --> 00:04:46.140
Or another way you could view
this is, is this is equal to--

00:04:46.140 --> 00:04:50.940
a percentage is just this times
100 so there's 0.8% chance of

00:04:50.940 --> 00:04:53.280
missing all of the free throws.

00:04:53.280 --> 00:04:57.120
Or another way of viewing this
is I have-- so what's my chance

00:04:57.120 --> 00:04:58.420
of making at least one?

00:04:58.420 --> 00:05:00.450
Well, it's going to
be 1 minus this.

00:05:00.450 --> 00:05:02.850
So what's 1 minus
0.-- this is a 0.

00:05:02.850 --> 00:05:03.370
0.008.

00:05:03.370 --> 00:05:09.320
Well it's 99.2%.

00:05:09.320 --> 00:05:14.470
Yeah, that's 99.2%.

00:05:14.470 --> 00:05:17.350
Or you could also
view it as 0.992.

00:05:17.350 --> 00:05:19.180
You take 1 minus this, you
get 0.992, which is the

00:05:19.180 --> 00:05:21.720
same thing as 99.2% chance.

00:05:21.720 --> 00:05:27.620
So I have a 99.2% chance of
at least tying the game.

00:05:27.620 --> 00:05:30.230
So it's pretty high if you have
someone at the free-throw line

00:05:30.230 --> 00:05:33.535
with a 80% free-throw
percentage and they

00:05:33.535 --> 00:05:34.300
have three shots.

00:05:34.300 --> 00:05:35.560
Now, what if I only
had two shots?

00:05:35.560 --> 00:05:37.280
Let's say I got fouled when
I was thinking a 2-pointer

00:05:37.280 --> 00:05:39.720
and I only have two shots.

00:05:39.720 --> 00:05:43.180
Well, in that case, in order to
tie the game I have to get at

00:05:43.180 --> 00:05:45.120
least one, but I only
have two shots.

00:05:45.120 --> 00:05:47.830
So it would be 1 minus
the chances of me

00:05:47.830 --> 00:05:49.660
missing both shots.

00:05:49.660 --> 00:05:52.090
What's the chances of
missing two shots?

00:05:52.090 --> 00:05:56.000
Well, the chance of missing
two shots in a row-- it's 4%.

00:05:56.000 --> 00:05:57.430
20% times 20%.

00:05:57.430 --> 00:06:06.240
It's a 4% chance of missing
two shots, two in a row.

00:06:06.240 --> 00:06:11.400
So my probability of getting at
least one, assuming that I am

00:06:11.400 --> 00:06:13.300
taking two shots, is going
to be 1 minus this.

00:06:13.300 --> 00:06:21.410
I have a 96% probability of at
least one shot if I take two.

00:06:21.410 --> 00:06:22.950
So that's also pretty high.

00:06:22.950 --> 00:06:24.170
And of course, what's
my probability if I

00:06:24.170 --> 00:06:25.730
only have one shot?

00:06:25.730 --> 00:06:27.560
Well, it's 80%.

00:06:27.560 --> 00:06:29.260
So hopefully that gives you a
little bit of framework next

00:06:29.260 --> 00:06:32.580
time you watch basketball game
and you can pause your TiVo and

00:06:32.580 --> 00:06:35.340
figure out the probability
when the person is making

00:06:35.340 --> 00:06:36.470
that last clutch shot.

00:06:36.470 --> 00:06:38.950
And it could be an interesting
experiment for you.

00:06:41.750 --> 00:06:44.380
And actually, I was thinking,
an interesting scientific

00:06:44.380 --> 00:06:48.280
experiment, or maybe a
high-school science project,

00:06:48.280 --> 00:06:50.980
people have a free-throw
percentage and that kind of

00:06:50.980 --> 00:06:54.550
implies that every time someone
takes a free throw that those

00:06:54.550 --> 00:06:57.090
are mutually exclusive events,
that they're independent of

00:06:57.090 --> 00:06:59.630
the previous time like
we said with coins.

00:06:59.630 --> 00:07:02.140
But an interesting idea, you
know in basketball people

00:07:02.140 --> 00:07:04.510
always say, he's hot now
or he has a streak.

00:07:04.510 --> 00:07:07.020
And so there is this notion and
I know I felt it, that there

00:07:07.020 --> 00:07:08.950
are times that you're
probability increases or

00:07:08.950 --> 00:07:12.370
decreases, and it tends to be
maybe dependent on whether you

00:07:12.370 --> 00:07:14.020
made or missed your
previous shot.

00:07:14.020 --> 00:07:16.170
So one thing you might want
to-- this is, I think, a

00:07:16.170 --> 00:07:20.430
legitimate science project-- is
to either get the data from

00:07:20.430 --> 00:07:25.120
real NBA players and see if
they really are mutually

00:07:25.120 --> 00:07:26.510
exclusive events.

00:07:26.510 --> 00:07:29.820
If the probability of making
the next shot or the next free

00:07:29.820 --> 00:07:32.680
throw really is independent of
whether they made or missed

00:07:32.680 --> 00:07:35.500
the previous one. or whether
it actually is dependent.

00:07:35.500 --> 00:07:38.320
Or if you don't have all the
data from the NBA or wherever,

00:07:38.320 --> 00:07:40.250
although I suspect you could
find it, you could try it with

00:07:40.250 --> 00:07:41.680
yourself and your friends.

00:07:41.680 --> 00:07:43.735
Or maybe since you want to
be unbiased, you'll do

00:07:43.735 --> 00:07:44.340
it with your friend.

00:07:44.340 --> 00:07:47.670
You'll see if the probability
of them making the next free

00:07:47.670 --> 00:07:51.920
throw really is independent of
whether they made the last one.

00:07:51.920 --> 00:07:54.250
Actually, I think that could be
quite good and you can get

00:07:54.250 --> 00:07:57.190
quite involved in the analysis.

00:07:57.190 --> 00:08:00.440
So let me finish this up
with another scenario.

00:08:00.440 --> 00:08:04.170
We talked about free throws, we
talked about flipping a coin,

00:08:04.170 --> 00:08:07.050
and now I'll talk about dice
because that is really another

00:08:07.050 --> 00:08:11.100
area where-- well, it's one
interesting and you'll probably

00:08:11.100 --> 00:08:14.930
see some problems
on probability.

00:08:14.930 --> 00:08:18.430
So, in general, when you're
playing games involving dice

00:08:18.430 --> 00:08:21.170
it's always interesting to say,
what's the probability if I

00:08:21.170 --> 00:08:24.470
have two six-sided dice, what's
my probability of getting--

00:08:24.470 --> 00:08:26.040
I don't know-- a
particular number.

00:08:26.040 --> 00:08:30.050
Let's say, what's the
probability of getting a 7?

00:08:30.050 --> 00:08:32.210
So to think about that you have
to say, well, what has to

00:08:32.210 --> 00:08:35.410
happen for the dice
for me to get a 7?

00:08:35.410 --> 00:08:37.020
And I think here it
might be interesting to

00:08:37.020 --> 00:08:38.680
draw a bit of a grid.

00:08:38.680 --> 00:08:41.660
So let's say that's my grid.

00:08:41.660 --> 00:08:44.340
Let me split it up into six.

00:08:44.340 --> 00:08:50.990
So let's see, that's splitting
it up into one, into two.

00:08:50.990 --> 00:08:52.785
Each of these maybe split
it up into three, so

00:08:52.785 --> 00:08:54.555
it would be like that.

00:08:54.555 --> 00:08:58.830
It won't be perfect, but close.

00:08:58.830 --> 00:08:59.335
Like that.

00:09:04.710 --> 00:09:05.160
Like that.

00:09:09.100 --> 00:09:09.840
And like that.

00:09:09.840 --> 00:09:12.150
And let me split it-- you'll
see what I'm doing in a second.

00:09:12.150 --> 00:09:12.990
Actually, I didn't
want to do that.

00:09:12.990 --> 00:09:14.560
That's good enough.

00:09:14.560 --> 00:09:17.420
Do this, this.

00:09:17.420 --> 00:09:18.560
I'm trying to make a 6-by-6.

00:09:23.830 --> 00:09:27.080
OK, and the reason why I'm
doing this is because let's

00:09:27.080 --> 00:09:32.350
make this top axis, this
horizontal-- each of the

00:09:32.350 --> 00:09:35.800
situations I can get on the
first dice, although I'm going

00:09:35.800 --> 00:09:37.670
to roll them simultaneously,
although it doesn't matter if I

00:09:37.670 --> 00:09:39.440
roll them simultaneously or one
after the other, or I roll one

00:09:39.440 --> 00:09:40.890
dice one after the other.

00:09:40.890 --> 00:09:46.690
So this first dice I could get
1, a 2, a 3, a 4, a 5, or a 6.

00:09:46.690 --> 00:09:49.310
So this is dice one-- D1.

00:09:49.310 --> 00:09:51.820
And on the second dice,
I could get a 1, a 2,

00:09:51.820 --> 00:09:54.410
a 3, a 4, a 5, or a 6.

00:09:54.410 --> 00:09:55.800
And this is dice two.

00:09:55.800 --> 00:09:57.170
And I'm running out of
time, so I will see

00:09:57.170 --> 00:09:58.910
you in the next video.


Probability (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.420
Let's keep doing some problems.

00:00:02.420 --> 00:00:07.630
So in the last time I said,
what are the chances of not

00:00:07.630 --> 00:00:12.480
getting any heads if I flip the
coin-- I was going to say dice,

00:00:12.480 --> 00:00:13.890
but I realized we're
dealing with a coin.

00:00:13.890 --> 00:00:15.580
If we flip the coin seven
times, if you said, well,

00:00:15.580 --> 00:00:18.200
that's the same thing as
getting 7 tails in a

00:00:18.200 --> 00:00:20.950
row, and that's 1/2
to the seventh power.

00:00:20.950 --> 00:00:25.980
Because each trial there's a
1/2 chance of getting the exact

00:00:25.980 --> 00:00:28.140
thing that we want, which
in this case is tails.

00:00:28.140 --> 00:00:30.210
And you multiply it times
itself seven times

00:00:30.210 --> 00:00:32.330
and you get 1/128.

00:00:32.330 --> 00:00:36.790
It actually turns out that
there are actually 1/128

00:00:36.790 --> 00:00:40.530
possible results that we can
get, equally probable results.

00:00:40.530 --> 00:00:41.680
And why do I say that?

00:00:41.680 --> 00:00:47.460
Well, what's the probability of
getting-- I don't know-- all

00:00:47.460 --> 00:00:50.600
heads and then the last
flip I get is tails?

00:00:50.600 --> 00:00:54.770
So heads, heads, heads,
heads-- so I'm doing it

00:00:54.770 --> 00:00:55.540
seven times, rights?

00:00:55.540 --> 00:01:02.630
So 1, 2, 3, 4, 5, 6, and then
the last time I get a tails.

00:01:02.630 --> 00:01:06.820
If you can think of it, this is
exactly 1 particular result out

00:01:06.820 --> 00:01:09.140
of the total results
that we can get.

00:01:09.140 --> 00:01:10.270
And what's the
probably of this?

00:01:10.270 --> 00:01:12.600
Well, there's a 1/2 chance of
getting a heads, 1/2 chance of

00:01:12.600 --> 00:01:15.760
a heads, 1/2 chance of heads--
let me switch colors-- 1/2

00:01:15.760 --> 00:01:18.720
chance of a heads, 1/2 chance
of a heads, 1/2 chance of a

00:01:18.720 --> 00:01:21.750
heads, and then 1/2
chance of a tails.

00:01:21.750 --> 00:01:24.650
Once again, this is just 1/2
times 1/2 times 1/2 times

00:01:24.650 --> 00:01:30.580
1/2 times 1/2 times 1/2,
which is equal to 1/128.

00:01:30.580 --> 00:01:35.690
So any particular-- I don't
want to say combination because

00:01:35.690 --> 00:01:37.910
we'll learn a little bit about
permutations and combinations

00:01:37.910 --> 00:01:39.080
in future videos.

00:01:39.080 --> 00:01:45.620
But any particular set of
circumstances we want is 1

00:01:45.620 --> 00:01:50.110
out of 128 of the total
number of outcomes.

00:01:50.110 --> 00:01:50.600
That what I should use.

00:01:50.600 --> 00:01:52.280
Instead of circumstances,
outcomes.

00:01:52.280 --> 00:01:54.490
If I have any particular
outcome-- so this is a

00:01:54.490 --> 00:01:59.090
particular outcome, the way to
view it is there are 128

00:01:59.090 --> 00:02:02.480
particular outcomes, so if I
choose one of them my odds of

00:02:02.480 --> 00:02:05.060
getting it is 1 out of 128.

00:02:05.060 --> 00:02:06.310
So let me ask you a question.

00:02:09.220 --> 00:02:17.470
What's the probability of
getting exactly 1 heads?

00:02:22.650 --> 00:02:24.230
So what does this mean?

00:02:24.230 --> 00:02:29.790
This means I could get-- oh, I
could get tails, tails, tails

00:02:29.790 --> 00:02:32.490
and-- let's say we're doing it
five times just so I don't

00:02:32.490 --> 00:02:33.610
waste too much time.

00:02:33.610 --> 00:02:35.655
Out of 5 rolls, out of 5 flips.

00:02:40.160 --> 00:02:45.130
So that could be
tails, tails, heads.

00:02:45.130 --> 00:02:47.290
That could be-- as you
can tell, I switched

00:02:47.290 --> 00:02:48.460
colors arbitrarily.

00:02:48.460 --> 00:02:54.320
That could be tails, tails,
tails, head, tails.

00:02:54.320 --> 00:02:55.210
I think you get the pattern.

00:02:55.210 --> 00:02:58.350
Tails, tails, heads,
tails, tails.

00:02:58.350 --> 00:03:02.120
That could be tails, heads,
tails, tails, tails.

00:03:02.120 --> 00:03:04.620
That could be tail-- oh sorry.

00:03:04.620 --> 00:03:08.270
That could be heads,
tails, tails, tails.

00:03:08.270 --> 00:03:12.160
So if we look at this way
there's this probability.

00:03:12.160 --> 00:03:15.280
There's actually how many
events, how many outcomes,

00:03:15.280 --> 00:03:17.730
would satisfy this
statement, exactly 1 heads?

00:03:17.730 --> 00:03:21.320
Well, there's 1, 2, 3, 4, 5.

00:03:21.320 --> 00:03:23.070
And the way you could think
of it is we're going to do

00:03:23.070 --> 00:03:26.840
5 flips, exactly 1 of the
flips is going to be heads.

00:03:26.840 --> 00:03:29.860
It could be any 1
of those 5 flips.

00:03:29.860 --> 00:03:32.050
There are 5 situations
in which that happens.

00:03:32.050 --> 00:03:33.860
And notice, I said
exactly one heads.

00:03:33.860 --> 00:03:35.750
Because if I said at least 1
heads then we would have to

00:03:35.750 --> 00:03:39.060
take into circumstances where
we have 2 heads and then it

00:03:39.060 --> 00:03:40.070
becomes more complicated.

00:03:40.070 --> 00:03:42.910
We'll learn more about that
when we do combinations.

00:03:42.910 --> 00:03:47.720
So what are the chances of any
one of these possible outcomes?

00:03:47.720 --> 00:03:50.890
Well, if we just
look at this one.

00:03:50.890 --> 00:03:54.830
There's a chance of 1/2,
1/2, 1/2, 1/2, 1/2, so it's

00:03:54.830 --> 00:03:57.610
1/2 to the fifth power.

00:03:57.610 --> 00:04:00.960
Which is equal to 1/32.

00:04:00.960 --> 00:04:02.770
And you use the same
logic each to these.

00:04:02.770 --> 00:04:12.500
This is 1/32, 1/32,
1/32, 1 out of 32.

00:04:12.500 --> 00:04:16.480
And in general, when I'm
flipping a fair coin five times

00:04:16.480 --> 00:04:20.510
in a row there are going to
be 32 possible outcomes.

00:04:20.510 --> 00:04:23.720
And each of these is just 1
of the particular outcome.

00:04:23.720 --> 00:04:26.980
And we're essentially saying,
out of the 32 outcomes, 5

00:04:26.980 --> 00:04:31.780
satisfy the event that we're
looking for-- exactly 1

00:04:31.780 --> 00:04:33.690
heads out of 5 flips.

00:04:33.690 --> 00:04:38.710
So there are 5 that satisfy it
out of a total of 32 outcomes.

00:04:38.710 --> 00:04:41.060
And so using that definition
only if we can say that all

00:04:41.060 --> 00:04:43.580
of the outcomes are
equally probable.

00:04:43.580 --> 00:04:49.200
The probability is 5/32.

00:04:49.200 --> 00:04:51.900
And then you could say, well,
let me make it a little

00:04:51.900 --> 00:04:53.820
bit more complicated.

00:04:53.820 --> 00:05:07.180
What is the probability of not
getting exactly one head?

00:05:07.180 --> 00:05:10.500
So here there is a lot more
circumstances that satisfy it.

00:05:10.500 --> 00:05:13.410
So for example, let me
throw out a couple

00:05:13.410 --> 00:05:14.285
that would satisfy it.

00:05:14.285 --> 00:05:15.090
Well, you could get all tails.

00:05:18.310 --> 00:05:20.865
But also, if you got 2 heads
that would satisfy it because

00:05:20.865 --> 00:05:25.780
you didn't get exactly 1 head,
So this would also satisfy it.

00:05:25.780 --> 00:05:27.300
And so you might say, well,
if we do it like this

00:05:27.300 --> 00:05:28.320
it's really complicated.

00:05:28.320 --> 00:05:36.440
Or you could say, well, these
5/32 are the only outcomes that

00:05:36.440 --> 00:05:40.580
don't satisfy-- that do not
satisfy this condition.

00:05:40.580 --> 00:05:43.590
So the rest of the what?

00:05:43.590 --> 00:05:44.800
32 minus 5 is what?

00:05:44.800 --> 00:05:45.660
27.

00:05:45.660 --> 00:05:49.390
The other 27 outcomes
will satisfy this.

00:05:49.390 --> 00:05:51.710
So you could say that this
is equal to-- and this is

00:05:51.710 --> 00:05:53.760
a common trick in a lot
of probability problems.

00:05:53.760 --> 00:06:02.130
This is equal to 1 minus the
probability of getting exactly

00:06:02.130 --> 00:06:04.325
one heads out of 5 flips.

00:06:07.280 --> 00:06:12.620
And that is equal to-- 1 is
the same thing as 32/32.

00:06:12.620 --> 00:06:14.220
Minus-- what's the
probability of this?

00:06:14.220 --> 00:06:14.960
We figured it out.

00:06:14.960 --> 00:06:17.120
It's 5/32.

00:06:17.120 --> 00:06:21.220
And that equals 27/32.

00:06:21.220 --> 00:06:23.090
So that's just always
something to keep in mind.

00:06:23.090 --> 00:06:26.820
Sometimes when you get a
probability problem it seems

00:06:26.820 --> 00:06:29.970
difficult to solve that
problem, but it's actually

00:06:29.970 --> 00:06:32.780
not that difficult to solve
the opposite problem.

00:06:32.780 --> 00:06:36.060
So you say, well, it's hard for
me to directly figure out the

00:06:36.060 --> 00:06:38.650
probability of not getting
exactly one heads; I'd have to

00:06:38.650 --> 00:06:40.170
know combinations
and all of this.

00:06:40.170 --> 00:06:44.430
But this is the opposite of
getting exactly one heads,

00:06:44.430 --> 00:06:46.860
and that's an easier thing
for me to figure out.

00:06:46.860 --> 00:06:49.230
So if I can figure out that
probability, all of the other

00:06:49.230 --> 00:06:51.930
outcomes would satisfy
the opposite one.

00:06:51.930 --> 00:06:53.830
Hope I didn't confuse you.

00:06:53.830 --> 00:06:56.160
So anyway, everything we've
done so far, it's been dealing

00:06:56.160 --> 00:07:00.530
with coins and fair coins and
that's interesting to a certain

00:07:00.530 --> 00:07:03.660
degree, but let's make it so
that the different outcomes

00:07:03.660 --> 00:07:06.380
have different probabilities.

00:07:06.380 --> 00:07:08.680
Let's do free throw percentages
because I think this is

00:07:08.680 --> 00:07:13.990
something that we all-- if we
watch basketball or play

00:07:13.990 --> 00:07:16.220
any basketball, we're
familiar with the idea.

00:07:16.220 --> 00:07:21.010
So let's say that I, and this
is not true, I only could

00:07:21.010 --> 00:07:25.720
wish-- let's say that I have
an 80% free throw percentage.

00:07:25.720 --> 00:07:30.940
So that means every time I go
to take a free throw there's an

00:07:30.940 --> 00:07:37.260
80% chance there's a basket and
that there's a 20% chance

00:07:37.260 --> 00:07:39.210
there's no basket.

00:07:39.210 --> 00:07:40.390
How does this come up with?

00:07:40.390 --> 00:07:42.780
Well, we'll do more on
statistics later and they're

00:07:42.780 --> 00:07:44.730
really-- probability and
statistics are opposite

00:07:44.730 --> 00:07:47.850
sides of the same I guess
coin, you could say.

00:07:47.850 --> 00:07:51.690
But someone says OK, out of the
last hundred times Sal took

00:07:51.690 --> 00:07:53.010
free throws he's made 80.

00:07:53.010 --> 00:07:57.420
So in general, he has an 80%
chance of making a particular

00:07:57.420 --> 00:07:58.720
free throw, but it might
not have been a hundred.

00:07:58.720 --> 00:08:01.840
I might have taken a thousand
free throws and I made 800.

00:08:01.840 --> 00:08:04.850
But that's where you come
up with the 80% from.

00:08:04.850 --> 00:08:07.270
So that's fair enough.

00:08:07.270 --> 00:08:12.745
So the 80% is p, and then
a common notation is

00:08:12.745 --> 00:08:16.080
the 20% is 1 minus p.

00:08:16.080 --> 00:08:20.020
If I have an 80% chance of
making the shot, I have a 20%

00:08:20.020 --> 00:08:22.990
chance of missing the shot.

00:08:22.990 --> 00:08:26.210
So my question to you, let's
do kind of the same examples

00:08:26.210 --> 00:08:27.620
we did with the coins.

00:08:27.620 --> 00:08:34.140
What's the probability of me
making-- I don't know-- 3 shots

00:08:34.140 --> 00:08:40.710
in a row, 3 baskets in a row,
or 3 free throws in a row?

00:08:40.710 --> 00:08:42.045
I'm going to say free throws
because this is my free

00:08:42.045 --> 00:08:45.130
throw percentage, not my
overall shot percentage.

00:08:45.130 --> 00:08:49.430
Well, by the same logic
it's going to be equal.

00:08:49.430 --> 00:08:51.910
The first one I would have to
get right, then the second one

00:08:51.910 --> 00:08:54.780
I would have to get-- I would
have to get a basket and

00:08:54.780 --> 00:08:57.460
there's a 20% I have no basket.

00:08:57.460 --> 00:09:00.240
And then the third one I
would have an 80% chance

00:09:00.240 --> 00:09:01.570
of getting a basket.

00:09:01.570 --> 00:09:04.060
So this is-- I didn't draw
the rest of the tree.

00:09:04.060 --> 00:09:06.430
It gets pretty broad.

00:09:06.430 --> 00:09:08.940
But in general, just from the
coin example you can just

00:09:08.940 --> 00:09:10.046
multiply these probabilities.

00:09:13.000 --> 00:09:16.460
So 80% is the same thing
as 0.8, so you get 0.8

00:09:16.460 --> 00:09:18.080
to the third power.

00:09:18.080 --> 00:09:19.780
I don't know, what is that?

00:09:19.780 --> 00:09:20.820
Let me see.

00:09:20.820 --> 00:09:30.940
It is 0.8 times 0.8 times
0.8 is equal to 51.2

00:09:30.940 --> 00:09:40.340
or 0.512-- 51.2%.

00:09:40.340 --> 00:09:43.880
So I have slightly better than
even odds of making 3 free

00:09:43.880 --> 00:09:46.790
throws in a row, which
is pretty good.

00:09:46.790 --> 00:09:49.030
And as you can see, I don't
have an exponent on this

00:09:49.030 --> 00:09:51.380
cheapo Windows provided
calculator I use.

00:09:51.380 --> 00:09:52.380
I have to apologize for that.

00:09:52.380 --> 00:09:54.465
So we might to limit
our exponent size.

00:09:54.465 --> 00:09:57.230
But anyway, that should
hopefully-- well, that's

00:09:57.230 --> 00:09:57.910
a little bit of a taste.

00:09:57.910 --> 00:10:00.830
In the next video I'll do a lot
more examples with the free

00:10:00.830 --> 00:10:02.420
throws and the basket, and
we'll even learn a little bit

00:10:02.420 --> 00:10:03.620
about conditional probability.

00:10:03.620 --> 00:10:05.270
See you soon.