How many cars can fit in the parking lot

WEBVTT
Kind: captions
Language: en

00:00:00.471 --> 00:00:02.637
The local grocery store opens at nine.

00:00:02.637 --> 00:00:04.809
Its parking lot has six rows.

00:00:04.809 --> 00:00:06.588
Each row can fit seven cars.

00:00:06.588 --> 00:00:08.285
Each car has four wheels.

00:00:08.317 --> 00:00:11.182
How many cars can the parking lot fit?

00:00:11.182 --> 00:00:13.960
And I encourage you to pause the video
and think about this yourself.

00:00:13.960 --> 00:00:16.641
Try to figure it out on your own.

00:00:16.641 --> 00:00:17.630
So, let's re-read this.

00:00:17.630 --> 00:00:19.890
The local grocery store opens up at nine.

00:00:19.890 --> 00:00:23.520
Well, that doesn't really matter. If we're thinking about
how many cars can the parking lot fit.

00:00:23.520 --> 00:00:25.495
So we don't really have to care about that.

00:00:25.495 --> 00:00:27.745
We also dont have to care about
how many wheels each car has.

00:00:27.745 --> 00:00:30.145
They're not asking us how many
wheels can fit in the parking lot.

00:00:30.145 --> 00:00:31.725
So we can ignore that.

00:00:31.725 --> 00:00:34.405
What we really care about is
how many rows we have.

00:00:34.405 --> 00:00:36.795
And how many cars can fit in each row.

00:00:36.795 --> 00:00:43.855
What we have is -- We have six rows
and each row can fit seven cars.

00:00:43.855 --> 00:00:47.255
We're going to six groups of seven.

00:00:47.255 --> 00:00:48.365
Or, another way of thinking about it.

00:00:48.365 --> 00:00:53.215
We're going to have six times seven
cars can fit in the parking lot.

00:00:53.215 --> 00:00:55.425
What is this going to be equal to?

00:00:55.425 --> 00:00:58.911
This is literally six sevens added up.

00:00:58.911 --> 00:01:05.451
This is the same thing as
one, two, three, four, five, six.

00:01:05.451 --> 00:01:07.951
That last seven looks strange.

00:01:07.951 --> 00:01:13.221
Now we're going to add these up.

00:01:13.221 --> 00:01:15.641
Seven plus seven is fourteen.

00:01:15.641 --> 00:01:20.401
Twenty-one, twenty-eight,
thirty-five, forty-two.

00:01:20.401 --> 00:01:25.141
Six times seven is equal to forty-two.

00:01:25.141 --> 00:01:31.021
So forty-two cars can fit in the parking lot.
Don't believe me? I made a little diagram here.

00:01:31.021 --> 00:01:32.731
We have six rows.

00:01:32.731 --> 00:01:34.071
This is the first row.

00:01:34.071 --> 00:01:34.901
Second..

00:01:34.901 --> 00:01:35.891
Third..

00:01:35.891 --> 00:01:36.751
Fourth..

00:01:36.751 --> 00:01:37.441
Fifth..

00:01:37.441 --> 00:01:38.861
Sixth.

00:01:38.861 --> 00:01:41.671
Each row can fit seven cars.
You see it here.

00:01:41.671 --> 00:01:44.071
One; Let me make that a little brighter.

00:01:44.071 --> 00:01:45.001
One..

00:01:45.001 --> 00:01:45.771
Two..

00:01:45.771 --> 00:01:46.741
Three..

00:01:46.741 --> 00:01:49.881
Four, five, six, seven.

00:01:49.881 --> 00:01:51.510
How many cars are there?

00:01:51.596 --> 00:01:53.053
You have seven.

00:01:53.053 --> 00:01:54.253
Fourteen..

00:01:54.253 --> 00:01:55.553
Twenty-one..

00:01:55.553 --> 00:01:56.793
Twenty-eight..

00:01:56.794 --> 00:01:57.934
Thirty-five..

00:01:57.934 --> 00:02:00.334
Forty-two total cars.

00:02:00.334 --> 00:02:04.334
Six rows of seven.


Negative numbers and number line examples

WEBVTT
Kind: captions
Language: en

00:00:00.852 --> 00:00:03.185
- We have three different
number lines here,

00:00:03.185 --> 00:00:04.730
and on each number line they mark off

00:00:04.730 --> 00:00:07.233
a couple of these marks,
this is negative two

00:00:07.233 --> 00:00:09.960
this is negative 10,
negative five, negative 11,

00:00:09.960 --> 00:00:12.985
and then we need to figure out
what the blue dot represents.

00:00:12.985 --> 00:00:14.473
And as a little bit of a hint,

00:00:14.473 --> 00:00:18.313
each mark here is not
necessarily incrementing by one,

00:00:18.313 --> 00:00:19.730
it could be by more than one.

00:00:19.730 --> 00:00:21.543
So I encourage you to pause this video,

00:00:21.543 --> 00:00:22.581
and to try it on your own.

00:00:22.581 --> 00:00:25.517
Try to figure out what number
does this blue dot represent

00:00:25.517 --> 00:00:26.883
on these different number lines?

00:00:28.039 --> 00:00:29.631
So let's tackle this first one.

00:00:29.631 --> 00:00:31.804
So we're gonna go where
they gave us this mark is

00:00:31.804 --> 00:00:33.335
negative two, this is negative 10,

00:00:33.335 --> 00:00:34.661
and we need to figure out this blue one

00:00:34.661 --> 00:00:37.649
that's further to the left of negative 10.

00:00:37.674 --> 00:00:41.470
Well just going from
negative two to negative 10,

00:00:41.470 --> 00:00:45.394
what has to happen to negative
two to get to negative 10?

00:00:45.953 --> 00:00:47.949
Well we'll have to subtract eight.

00:00:47.949 --> 00:00:49.734
And so if we move two to the left,

00:00:49.734 --> 00:00:51.798
that's the equivalent
of subtracting eight.

00:00:51.798 --> 00:00:53.797
So if we move two to the left again,

00:00:53.797 --> 00:00:55.709
that's subtracting eight again.

00:00:55.709 --> 00:00:57.989
So this must be negative 18

00:00:57.989 --> 00:01:00.062
and so if we move two to the left again,

00:01:00.062 --> 00:01:02.562
that also must be subtracting eight.

00:01:02.587 --> 00:01:07.587
Negative 18 minus eight,
would get us to negative,

00:01:08.072 --> 00:01:11.995
that gets us to negative 26.

00:01:12.894 --> 00:01:14.821
Now another way we could
have thought about it is,

00:01:14.821 --> 00:01:17.533
if we jump two to the left
and that's negative eight,

00:01:17.533 --> 00:01:20.222
then one jump to the left
is gonna be negative four.

00:01:20.222 --> 00:01:22.245
So you could say "Well
this is negative four",

00:01:22.245 --> 00:01:24.125
that would get you to negative six,

00:01:24.125 --> 00:01:27.213
then negative four again
gets you to negative 10,

00:01:27.213 --> 00:01:30.325
then this would be
negative 14, negative 18,

00:01:30.325 --> 00:01:34.089
negative 22, negative 26.

00:01:35.189 --> 00:01:36.653
Now let's tackle this one.

00:01:36.653 --> 00:01:38.141
So here we're gonna, we have to figure out

00:01:38.141 --> 00:01:39.924
what this blue dot here is on the right.

00:01:39.924 --> 00:01:44.924
So if we started at 11, you make two jumps

00:01:44.950 --> 00:01:47.690
to go to negative five.

00:01:47.690 --> 00:01:49.842
So what do I have to add to negative 11

00:01:49.842 --> 00:01:51.562
to get to negative five?

00:01:51.562 --> 00:01:53.746
Well I have to add six.

00:01:53.746 --> 00:01:55.890
So if I add six over two jumps,

00:01:55.890 --> 00:01:57.370
that means that each of these jumps,

00:01:57.370 --> 00:01:59.034
that must be plus three,

00:01:59.034 --> 00:02:01.554
and this one must be plus three.

00:02:01.554 --> 00:02:04.529
So now each mark we just add three.

00:02:04.529 --> 00:02:07.379
So negative five plus three...

00:02:07.404 --> 00:02:10.836
So plus three is gonna
get us to negative two.

00:02:10.861 --> 00:02:15.084
Plus three gets us to positive one,

00:02:15.084 --> 00:02:17.190
negative two plus three is positive one,

00:02:17.190 --> 00:02:20.714
plus three gets us to four.

00:02:20.714 --> 00:02:22.498
So here we got to negative 26,

00:02:22.498 --> 00:02:23.969
here we get to four.

00:02:23.969 --> 00:02:26.017
Now let's tackle this one.

00:02:26.510 --> 00:02:28.242
So we're at negative seven,

00:02:28.242 --> 00:02:30.057
or this is negative seven this is one

00:02:30.057 --> 00:02:32.297
and then we have to figure
out this mark right over here.

00:02:32.297 --> 00:02:34.001
So if you take two jumps to the right,

00:02:34.001 --> 00:02:35.459
from negative seven to one,

00:02:35.459 --> 00:02:37.097
how much did we have to add?

00:02:37.368 --> 00:02:39.376
Well to get from negative seven to one,

00:02:39.376 --> 00:02:41.157
you have to add eight.

00:02:41.252 --> 00:02:42.418
You have to add eight.

00:02:42.443 --> 00:02:45.530
So then if we make another two
jumps to the mystery number

00:02:45.530 --> 00:02:47.402
that means we added eight again.

00:02:47.402 --> 00:02:50.399
So one plus eight is nine.

00:02:50.399 --> 00:02:52.495
Another way we could
have thought about it,

00:02:52.495 --> 00:02:54.487
if we added eight over two jumps

00:02:54.487 --> 00:02:57.075
that means we added four on each jump.

00:02:57.100 --> 00:02:59.919
So this one must be negative three,

00:02:59.919 --> 00:03:01.750
add four you get to one,

00:03:01.750 --> 00:03:04.237
add four you get to five,

00:03:04.237 --> 00:03:06.992
add four you get to nine.


Free throw binomial probability distribution

WEBVTT
Kind: captions
Language: en

00:00:00.015 --> 00:00:02.014
- [Voiceover] Now that we've
spent a couple of videos

00:00:02.014 --> 00:00:05.195
exploring a scenario where I'm
taking multiple free throws

00:00:05.195 --> 00:00:07.516
and figuring out the
probability of making K

00:00:07.516 --> 00:00:10.395
of the scores and six
attempts or in N attempts.

00:00:10.395 --> 00:00:12.054
Let's actually define a random variable

00:00:12.054 --> 00:00:14.355
using this scenario and see if we can

00:00:14.355 --> 00:00:16.635
construct it's probability distribution

00:00:16.635 --> 00:00:19.756
and we'll actually see that
it's a binomial distribution.

00:00:19.756 --> 00:00:22.724
So, let's define the random variable X.

00:00:22.724 --> 00:00:26.644
So, let's say that X
is equal to the number,

00:00:26.644 --> 00:00:30.254
the number of made shots,

00:00:30.254 --> 00:00:33.226
number of made free throws

00:00:33.226 --> 00:00:36.884
when taking,

00:00:36.884 --> 00:00:39.425
when taking six free throws.

00:00:39.425 --> 00:00:41.339
So, it's how many of the six do you make?

00:00:41.339 --> 00:00:42.859
And we're going to assume what we assumed

00:00:42.859 --> 00:00:44.899
in the first video in this series of this,

00:00:44.899 --> 00:00:46.700
these making free throws.

00:00:46.700 --> 00:00:48.757
So, we're gonna assume the seventy percent

00:00:48.757 --> 00:00:51.433
free throw probability right over here.

00:00:51.433 --> 00:00:53.894
So, assuming assumptions,

00:00:53.894 --> 00:00:58.894
assuming seventy percent free throw,

00:00:59.072 --> 00:01:01.894
free throw percentage.

00:01:01.894 --> 00:01:02.894
All right.

00:01:02.894 --> 00:01:04.272
So, let's figure out the probabilities

00:01:04.272 --> 00:01:07.793
of the different values that
X could actually take on.

00:01:07.793 --> 00:01:10.175
So, let's see, what is the probability,

00:01:10.175 --> 00:01:14.655
what is the probability
that X is equal to zero?

00:01:14.655 --> 00:01:15.348
That even though you have a

00:01:15.348 --> 00:01:18.247
seventy percent free throw percentage

00:01:18.247 --> 00:01:19.801
that you make none of the shots

00:01:19.801 --> 00:01:22.021
and actually you could
calculate this through

00:01:22.021 --> 00:01:24.356
probably some common
sense without using all

00:01:24.356 --> 00:01:25.999
of these fancy things but just

00:01:25.999 --> 00:01:27.571
to make things consistent,
I'm gonna write it out.

00:01:27.571 --> 00:01:32.110
So, this is going to
be, this is going to be,

00:01:32.110 --> 00:01:37.110
it's going to be equal to six choose zero

00:01:37.353 --> 00:01:42.353
times zero point seven to the zeroth power

00:01:42.490 --> 00:01:47.361
times zero point three to the sixth power,

00:01:49.553 --> 00:01:52.891
and this right over here
is gonna end up being one.

00:01:52.891 --> 00:01:55.170
This over here's going
to end up being one.

00:01:55.170 --> 00:01:57.851
And so, you're just gonna be
left with zero point three

00:01:57.851 --> 00:02:02.785
to the sixth power and I
calculated it ahead of time.

00:02:02.785 --> 00:02:05.205
So, if we round to the nearest,

00:02:05.205 --> 00:02:08.624
if we do, if we round our
percentages to the nearest tenth,

00:02:08.624 --> 00:02:11.125
this is going to give you approximately,

00:02:11.125 --> 00:02:15.260
approximately, well, if we round
the decimal to the nearest,

00:02:15.260 --> 00:02:17.556
to the nearest thousandth you're gonna

00:02:17.556 --> 00:02:19.778
get something like that
which is approximately

00:02:19.778 --> 00:02:23.978
equal to zero point one percent chance

00:02:23.978 --> 00:02:26.472
of you missing all of them.

00:02:26.472 --> 00:02:28.394
So, roughly I'm speaking, roughly here

00:02:28.394 --> 00:02:31.853
a one in a thousand,
one in a thousand chance

00:02:31.853 --> 00:02:35.034
of that happening, of
missing all six free throws.

00:02:35.034 --> 00:02:37.655
Now, let's keep going, this is fun.

00:02:37.655 --> 00:02:40.674
So, what is the probability that our

00:02:40.674 --> 00:02:43.074
random variable is equal to 1?

00:02:43.074 --> 00:02:45.873
Well, this is going to be six choose one

00:02:45.873 --> 00:02:49.331
times zero point seven to the first power

00:02:49.331 --> 00:02:54.267
times zero point three to
the six minus first power.

00:02:56.265 --> 00:02:57.328
So, it's a fifth power.

00:02:57.328 --> 00:02:59.407
And I calculated this and this is

00:02:59.407 --> 00:03:02.967
approximately zero point zero one

00:03:02.967 --> 00:03:05.462
or we could say one percent.

00:03:05.462 --> 00:03:06.715
So, it's still a fairly low probability.

00:03:06.715 --> 00:03:08.635
Ten times more likely than this roughly

00:03:08.635 --> 00:03:11.173
but still a fairly low probability.

00:03:11.173 --> 00:03:12.416
Let's keep going.

00:03:12.416 --> 00:03:15.329
So, the probability
that x is equal to two,

00:03:15.329 --> 00:03:18.950
well, that's what our first
video was essentially.

00:03:18.950 --> 00:03:21.789
So, this is going to be six choose two

00:03:21.789 --> 00:03:26.789
times zero point seven squared times zero,

00:03:27.350 --> 00:03:31.441
zero point three to the fourth power,

00:03:31.441 --> 00:03:35.858
and we saw that this is
approximately going to be

00:03:35.858 --> 00:03:40.858
zero, zero point zero six,
or we could say six percent,

00:03:42.385 --> 00:03:44.579
and obviously you could type
these things in a calculator

00:03:44.579 --> 00:03:46.314
and get a much more precise answer

00:03:46.314 --> 00:03:48.713
but just for the sake
of just getting a sense

00:03:48.713 --> 00:03:50.416
of what these probabilities all look like,

00:03:50.416 --> 00:03:52.450
that's why I'm giving
these rough estimates.

00:03:52.450 --> 00:03:55.970
Kind of, I guess you
could say, to the closest,

00:03:55.970 --> 00:03:58.423
maybe, tenth of a, tenth of a percent.

00:03:58.423 --> 00:04:00.343
And actually and if you
round to the closest tenth

00:04:00.343 --> 00:04:01.623
of a percent you actually actually get

00:04:01.623 --> 00:04:06.024
to six point oh percent
and this is one point,

00:04:06.024 --> 00:04:08.541
one point oh percent 'cause
this we actually went

00:04:08.541 --> 00:04:10.480
to a tenth of a percent
here but let's keep going.

00:04:10.480 --> 00:04:13.854
We're obviously going to have
to do a few more of these.

00:04:13.854 --> 00:04:16.769
So, let me just make sure
I have enough real estate.

00:04:16.785 --> 00:04:19.932
All right, so, the probability

00:04:19.932 --> 00:04:22.538
that our random variable is equal to three

00:04:22.538 --> 00:04:24.834
is going to be six choosethree

00:04:24.834 --> 00:04:27.191
and I'm sure you could probably
fill this out on your own

00:04:27.191 --> 00:04:28.712
but I'm going to do it.

00:04:28.712 --> 00:04:31.827
Zero point seven to the third power

00:04:31.827 --> 00:04:35.288
times zero point three
to the six minus three

00:04:35.288 --> 00:04:39.697
which is a third power which
is approximately equal to,

00:04:39.697 --> 00:04:43.902
well, it's going to be
zero point one eight five

00:04:43.902 --> 00:04:48.902
or eighteen point five,
eighteen point five percent.

00:04:49.556 --> 00:04:52.654
So, yeah, that's definitely
within the realm of possibility.

00:04:52.654 --> 00:04:54.095
I mean, all of these are
in the realm of possibility

00:04:54.095 --> 00:04:58.218
but it's starting to be a
non-insignificant probability.

00:04:58.218 --> 00:05:00.596
So, now, let's do the probability

00:05:00.596 --> 00:05:03.479
that our random variable is equal to four.

00:05:03.479 --> 00:05:05.599
Well, it's going to be six choose four

00:05:05.599 --> 00:05:09.042
times zero point seven to the fourth power

00:05:09.042 --> 00:05:11.450
times zero point three
to the six minus four

00:05:11.450 --> 00:05:14.250
or second power which is equal to,

00:05:14.250 --> 00:05:16.285
this is going to get equal to

00:05:16.285 --> 00:05:18.586
or approximately, 'cause
I am, I am taking away

00:05:18.586 --> 00:05:19.491
a little bit of the precision when

00:05:19.491 --> 00:05:21.275
I write these things down.

00:05:21.275 --> 00:05:26.275
Zero point three two four.

00:05:26.446 --> 00:05:31.295
So, approximately thirty two
point four percent chance

00:05:32.475 --> 00:05:36.105
of making exactly four out
of the six free throws.

00:05:36.105 --> 00:05:37.786
All right, two more to go.

00:05:37.786 --> 00:05:40.324
Let's see, I have not used purple as yet.

00:05:40.324 --> 00:05:44.815
So, the probability that a
random variable is equal to five

00:05:44.815 --> 00:05:48.515
it's gonna be six choose five or zero,

00:05:48.515 --> 00:05:51.531
at times I should say, zero
point seven to the fifth power

00:05:51.531 --> 00:05:55.894
times zero point three to the first power

00:05:55.894 --> 00:05:59.974
and that is going to be roughly,

00:05:59.974 --> 00:06:04.535
roughly zero point three zero three

00:06:04.535 --> 00:06:09.413
which is thirty point three percent.

00:06:09.413 --> 00:06:11.355
That's interesting, one more left.

00:06:11.355 --> 00:06:15.263
So, the probability
that I make all of them,

00:06:15.263 --> 00:06:18.333
of all six, is going to be equal to,

00:06:18.333 --> 00:06:22.533
is equal to six choose six

00:06:22.533 --> 00:06:25.554
and zero point seven to the sixth power

00:06:25.554 --> 00:06:28.626
times zero point three to the zeroth power

00:06:28.626 --> 00:06:30.933
which is, this right over
here is going to be one,

00:06:30.933 --> 00:06:32.275
this is going to be one, so it's really

00:06:32.275 --> 00:06:36.333
just zero point seven to the
sixth, to the sixth power

00:06:36.333 --> 00:06:41.333
and this is approximately
zero point one one eight.

00:06:42.534 --> 00:06:44.194
I calculated that ahead of time,

00:06:44.194 --> 00:06:47.635
which is eleven point eight.

00:06:47.635 --> 00:06:49.870
Eleven point eight percent.

00:06:49.870 --> 00:06:51.523
And so, there's something
interesting that's going on here.

00:06:51.523 --> 00:06:53.402
The first time we looked
binomial distribution

00:06:53.402 --> 00:06:55.682
we said, "Hey, there's
a symmetry as we kind of

00:06:55.682 --> 00:06:57.464
"got to some type of a peak and went down,

00:06:57.464 --> 00:06:58.984
"but I don't see that symmetry here."

00:06:58.984 --> 00:07:00.824
And the reason you're not
seeing that symmetry is

00:07:00.824 --> 00:07:03.352
that you are more likely to
make a free throw than not.

00:07:03.352 --> 00:07:05.944
It was you have a seventy
percent free throw probability.

00:07:05.944 --> 00:07:08.593
This is no longer just
flipping a fair coin.

00:07:08.609 --> 00:07:11.691
Where you will see the symmetry
is in these coefficients.

00:07:11.691 --> 00:07:13.531
If you calculate these coefficients,

00:07:13.531 --> 00:07:16.272
six choose zero is one.

00:07:16.272 --> 00:07:18.713
Six choose six is one.

00:07:18.713 --> 00:07:22.750
You would see that six choose one is six

00:07:22.750 --> 00:07:25.290
and six choose five is six.

00:07:25.290 --> 00:07:28.778
You'd see six choose two is fifteen

00:07:28.778 --> 00:07:32.538
and six choose four is also fifteen,

00:07:32.538 --> 00:07:35.587
and then six choose three is twenty.

00:07:35.587 --> 00:07:37.907
So, you definitely see, you definitely see

00:07:37.907 --> 00:07:39.427
the symmetry in the coefficients

00:07:39.427 --> 00:07:41.206
but then these things are weighted

00:07:41.206 --> 00:07:42.404
by the fact that you're more likely

00:07:42.404 --> 00:07:44.064
to make something than miss something.

00:07:44.064 --> 00:07:45.602
If these were both point five

00:07:45.602 --> 00:07:49.065
then you would also see the
symmetry right over here.

00:07:49.065 --> 00:07:51.842
And you can plot this
to essentially visualize

00:07:51.842 --> 00:07:54.344
what the probability distribution

00:07:54.344 --> 00:07:55.981
looks like for this example

00:07:55.981 --> 00:07:57.519
and I encourage you to do that.

00:07:57.519 --> 00:07:59.238
To take these different cases,

00:07:59.238 --> 00:08:00.617
just like we did in that first example

00:08:00.617 --> 00:08:03.278
with the fair coin, and plot these.

00:08:03.278 --> 00:08:04.538
But this essentially does give you

00:08:04.538 --> 00:08:07.245
the probability distribution for,

00:08:07.276 --> 00:08:08.575
for the random variable in question.

00:08:08.575 --> 00:08:11.135
This is, I just wrote it out
instead of just visualizing it

00:08:11.135 --> 00:08:12.592
but it says, "Okay, well, what's the...

00:08:12.592 --> 00:08:14.369
"These are the different values

00:08:14.369 --> 00:08:16.128
"that this random variable can take on."

00:08:16.128 --> 00:08:18.068
It can't, it can't take on negative one

00:08:18.068 --> 00:08:22.647
or it can't be fifteen point
five or pi or one million.

00:08:22.647 --> 00:08:25.446
These are the only seven values
that this random variable

00:08:25.446 --> 00:08:28.526
can take on and I've just
given you the probabilities,

00:08:28.526 --> 00:08:30.006
or I guess you could say
the rough probabilities

00:08:30.006 --> 00:08:32.327
of the random variable taking on

00:08:32.327 --> 00:08:37.202
each of those, each of those seven values.


Graphing basketball binomial distribution

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:02.319
- In the last video, we worked through

00:00:02.319 --> 00:00:03.966
essentially the probability distribution

00:00:03.966 --> 00:00:06.683
for this random variable defined as

00:00:06.683 --> 00:00:08.378
the number of free throws you make

00:00:08.378 --> 00:00:10.177
when taking size free throws,

00:00:10.177 --> 00:00:12.777
assuming you have a 70
percent free throw percentage,

00:00:12.777 --> 00:00:15.041
and I suggested, hey, why
don't you visualize this?

00:00:15.041 --> 00:00:18.106
Draw the graph of this
probability distribution,

00:00:18.106 --> 00:00:20.405
this binomial probability distribution.

00:00:20.405 --> 00:00:21.519
And when I thought about it, I said,

00:00:21.519 --> 00:00:23.557
well, I too would enjoy graphing it,

00:00:23.557 --> 00:00:24.708
and we might as well do it together,

00:00:24.708 --> 00:00:26.541
because whenever you graph these things

00:00:26.541 --> 00:00:27.563
it makes it very visual,

00:00:27.563 --> 00:00:31.765
and kind of the shape of a
binomial distribution like this.

00:00:31.765 --> 00:00:33.614
So let's do that.

00:00:33.614 --> 00:00:37.132
So let me maybe move over
to the right a little bit,

00:00:37.132 --> 00:00:38.177
I really just need to
be able to keep track

00:00:38.177 --> 00:00:42.036
of these things right over here.

00:00:42.036 --> 00:00:45.421
Let me draw some lines.

00:00:45.605 --> 00:00:49.983
If I were to just draw one line there,

00:00:49.983 --> 00:00:53.919
and then another line here,

00:00:53.919 --> 00:00:57.064
and then we have the
different percentages.

00:00:57.064 --> 00:00:58.539
So let's do that.

00:00:58.539 --> 00:01:02.359
See, the highest one is
a little over 32 percent.

00:01:02.359 --> 00:01:05.842
So maybe we'll go as
high as 40 percent here.

00:01:05.842 --> 00:01:10.842
40 percent, and then
this would be 20 percent.

00:01:12.077 --> 00:01:15.177
20, that looks about half way, 20 percent.

00:01:15.177 --> 00:01:19.680
This would be 10 percent.

00:01:19.680 --> 00:01:23.918
And this would be 30 percent.

00:01:23.918 --> 00:01:27.075
And then in this axis, let's
do the different values

00:01:27.075 --> 00:01:29.583
that the random variable could take on.

00:01:29.583 --> 00:01:33.090
So the random variable taking on

00:01:33.090 --> 00:01:35.741
the value zero.

00:01:35.741 --> 00:01:40.280
The random variable
taking on the value one.

00:01:40.280 --> 00:01:45.280
Zero, one, the random
variable taking on two.

00:01:46.604 --> 00:01:49.621
Two, we're almost there.

00:01:49.621 --> 00:01:54.621
Let's see, three, and then four.

00:01:55.170 --> 00:01:59.687
Four, and then five.

00:01:59.687 --> 00:02:02.729
Five, and then finally six.

00:02:02.729 --> 00:02:05.005
Take x equals six.

00:02:05.005 --> 00:02:07.626
And then six, and now let's
just graph all of these.

00:02:07.626 --> 00:02:10.712
So this first one, zero point one percent,

00:02:10.712 --> 00:02:11.851
well, that's barely going to register

00:02:11.851 --> 00:02:13.093
on this graph right here,

00:02:13.093 --> 00:02:14.787
so I'll just kind of
just give it a little bit

00:02:14.787 --> 00:02:16.923
of a showing right over there.

00:02:16.923 --> 00:02:19.153
Showing (mumbles) due to that green color.

00:02:19.153 --> 00:02:21.800
So, let me make sure.

00:02:21.800 --> 00:02:23.459
So, in that green color,

00:02:23.459 --> 00:02:25.874
you're going to have just
a little bit of a showing.

00:02:25.874 --> 00:02:28.579
One, as well, is kind of barely a showing.

00:02:28.579 --> 00:02:31.790
So it shows up a little bit more.

00:02:31.790 --> 00:02:34.957
So let me draw it like that.

00:02:34.957 --> 00:02:37.547
That is one percent, right over there.

00:02:37.547 --> 00:02:40.542
Now, two is six percent,
which on this scale,

00:02:40.542 --> 00:02:44.664
is going to be about that high.

00:02:44.664 --> 00:02:46.522
So let me draw it like that.

00:02:46.522 --> 00:02:48.728
So that is two.

00:02:48.728 --> 00:02:52.301
So that is six percent right there.

00:02:52.301 --> 00:02:55.173
X equaling three, 18
and a half percent shot

00:02:55.173 --> 00:02:56.646
of that happening...

00:02:56.646 --> 00:03:01.646
So 18 and a half, it's a hand drawn chart,

00:03:04.793 --> 00:03:07.585
or histogram, so you have to bear with me.

00:03:07.585 --> 00:03:12.585
So it's roughly there, and then four

00:03:14.928 --> 00:03:19.928
was 32.4 percent, so that is up here.

00:03:20.907 --> 00:03:25.907
So 32.4 percent looks like that.

00:03:28.894 --> 00:03:33.491
Let me shade that in, 32.4 percent.

00:03:33.491 --> 00:03:36.846
And then five was 30.3 percent,

00:03:36.846 --> 00:03:41.793
30.3 percent, slightly
lower, just like that.

00:03:41.793 --> 00:03:46.251
And it will look like this.

00:03:46.251 --> 00:03:51.251
30.3 percent, and finally,
six is 11.8 percent.

00:03:52.635 --> 00:03:53.726
So really, this whole video was just

00:03:53.726 --> 00:03:56.768
an exercise in making a histogram,

00:03:56.768 --> 00:04:01.156
but it's useful, because
to actually visualize

00:04:01.156 --> 00:04:03.055
what the distribution looks like.

00:04:03.055 --> 00:04:04.101
And what's really interesting,

00:04:04.101 --> 00:04:05.934
is to think about how does this change

00:04:05.934 --> 00:04:09.358
as you change the free throw percentage,

00:04:09.358 --> 00:04:11.901
or as you change the
number of shots you take,

00:04:11.901 --> 00:04:16.255
how does this change this
binomial distribution?

00:04:16.255 --> 00:04:17.241
And you can do that on a spreadsheet,

00:04:17.241 --> 00:04:21.757
and actually see how that all works out.


Generalizing k scores in n attempts

WEBVTT
Kind: captions
Language: en

00:00:00.467 --> 00:00:01.600
- [Voiceover] So the last video,

00:00:01.600 --> 00:00:03.837
we studied the circumstance

00:00:03.837 --> 00:00:06.466
where I had a 70% free throw probability.

00:00:06.466 --> 00:00:08.966
I have a 70% chance of
making any free throw,

00:00:08.966 --> 00:00:10.033
which is actually higher

00:00:10.033 --> 00:00:11.933
than my actual free throw probability,

00:00:11.933 --> 00:00:13.533
which might be a surprise to you.

00:00:13.533 --> 00:00:14.766
But we said in that circumstance,

00:00:14.766 --> 00:00:16.233
if it's a 70% chance of making it,

00:00:16.233 --> 00:00:18.369
well, that means that
you have a one minus 70%,

00:00:18.369 --> 00:00:20.233
or 30%, chance of missing.

00:00:20.400 --> 00:00:22.433
And we said if you took six attempts,

00:00:22.433 --> 00:00:24.400
the probability of you getting exactly,

00:00:24.400 --> 00:00:25.866
making two of the baskets,

00:00:25.866 --> 00:00:27.333
exactly two scores ...

00:00:27.333 --> 00:00:28.833
I called them "scores"
instead of "making it,"

00:00:28.833 --> 00:00:30.533
just because I wanted "making" and "miss"

00:00:30.533 --> 00:00:32.534
to have different
letters in that video ...

00:00:32.534 --> 00:00:36.933
We said, well, there's six
choose two different ways

00:00:36.933 --> 00:00:41.067
of making two, exactly two
out of the six free throws,

00:00:41.067 --> 00:00:43.633
and then the probability
of any one of those ways

00:00:43.633 --> 00:00:45.566
is going to be making it twice,

00:00:45.566 --> 00:00:47.433
which is 0.7 squared,

00:00:47.433 --> 00:00:49.900
and missing it four times,

00:00:49.900 --> 00:00:52.266
so 0.3 to the fourth power.

00:00:52.711 --> 00:00:54.869
This was just one particular situation,

00:00:54.869 --> 00:00:56.133
but we could generalize

00:00:56.133 --> 00:00:58.200
based on the logic that
we had in that video.

00:00:58.200 --> 00:00:59.433
In fact, let's do that.

00:00:59.433 --> 00:01:01.168
So if I were to generalize it,

00:01:01.168 --> 00:01:03.233
if I were to say the probability,

00:01:04.000 --> 00:01:05.168
the probability ...

00:01:05.168 --> 00:01:06.566
It's the exact same logic

00:01:06.566 --> 00:01:08.066
of exactly ...

00:01:08.966 --> 00:01:10.300
Exactly ...

00:01:10.966 --> 00:01:12.433
Now let's say k ...

00:01:13.200 --> 00:01:15.033
Let me do this in a color,
an interesting color.

00:01:15.033 --> 00:01:18.000
So let me do it in this
orangeish-brown color.

00:01:18.000 --> 00:01:21.367
K shots,

00:01:21.433 --> 00:01:23.300
or exactly k scores ...

00:01:23.300 --> 00:01:25.133
I'll call making a free throw a score.

00:01:25.133 --> 00:01:26.733
We'll just assume you got a point for it.

00:01:26.733 --> 00:01:29.133
So exactly two k scores ...

00:01:33.286 --> 00:01:35.833
in n attempts ...

00:01:35.969 --> 00:01:37.801
Let's just say in n attempts.

00:01:38.600 --> 00:01:42.135
In n, and let me go back
to that green color.

00:01:42.333 --> 00:01:44.267
N attempts ...

00:01:45.574 --> 00:01:47.975
N attempts

00:01:49.266 --> 00:01:51.368
is going to be equal to ...

00:01:52.100 --> 00:01:55.533
Well, how many ways can
you pick k things out of n,

00:01:55.533 --> 00:01:56.933
or n choose k?

00:01:57.966 --> 00:02:00.366
N choose k ...

00:02:00.733 --> 00:02:03.768
N choose k.

00:02:04.617 --> 00:02:06.200
Actually, let's just
generalize it even more.

00:02:06.200 --> 00:02:07.533
Let's just say that you have,

00:02:07.533 --> 00:02:09.966
your free throw probability is p.

00:02:10.200 --> 00:02:13.500
So let's say p is ...

00:02:13.500 --> 00:02:14.770
So for this situation right over here,

00:02:14.770 --> 00:02:16.233
since we generalized it fully,

00:02:16.233 --> 00:02:20.633
let's say that p is the
probability of making a free throw.

00:02:20.633 --> 00:02:21.933
Actually, since I already have a p here,

00:02:21.933 --> 00:02:25.466
let me just say f is
equal to the probability

00:02:26.266 --> 00:02:28.033
of making

00:02:29.357 --> 00:02:30.733
a free throw.

00:02:30.966 --> 00:02:32.666
Or you could say, your
probability of scoring,

00:02:32.666 --> 00:02:34.966
if you call a score making a free throw.

00:02:35.067 --> 00:02:37.167
So if f is your probability
of making a free throw ...

00:02:37.333 --> 00:02:38.866
So if you want n scores,

00:02:39.000 --> 00:02:40.333
then this is going to be ...

00:02:41.233 --> 00:02:42.433
This is going to be ...

00:02:42.566 --> 00:02:45.499
Well, it's going to be f to the n power,

00:02:45.766 --> 00:02:47.133
and then you're going to have ...

00:02:47.133 --> 00:02:49.000
and then you're going to
miss the remainder ...

00:02:49.166 --> 00:02:50.766
Sorry, f to the k power,

00:02:51.100 --> 00:02:53.233
because you're making exactly k scores.

00:02:53.669 --> 00:02:57.333
So f to the k power,

00:02:57.602 --> 00:02:58.933
and then the remainder,

00:02:58.933 --> 00:03:00.900
so the n minus k attempts,

00:03:00.900 --> 00:03:02.066
you're going to miss it.

00:03:02.167 --> 00:03:04.333
It's going to be that
probability of missing,

00:03:04.333 --> 00:03:07.433
and the probability of missing
is going to be one minus f,

00:03:07.868 --> 00:03:08.824
so it's going to be

00:03:09.347 --> 00:03:12.700
times one minus f

00:03:13.133 --> 00:03:15.967
to the n minus k power.

00:03:16.533 --> 00:03:20.333
To the n minus k power.

00:03:20.900 --> 00:03:23.400
And just, if you like, or I encourage you,

00:03:23.400 --> 00:03:24.200
pause the video

00:03:24.200 --> 00:03:25.866
and just make sure you
understand the parallels

00:03:25.866 --> 00:03:28.411
between this example where I had ...

00:03:29.119 --> 00:03:32.633
f was 70%, our f was 70%.

00:03:32.633 --> 00:03:33.800
One minus f ...

00:03:33.800 --> 00:03:37.366
Or f was .7 and one minus f would be .3,

00:03:37.567 --> 00:03:38.601
and we were seeing,

00:03:38.601 --> 00:03:42.167
how do we get two scores in six attempts?

00:03:42.167 --> 00:03:44.868
And here we're saying,
k scores in n attempts.

00:03:45.066 --> 00:03:47.767
This is just a general
way to think about it.

00:03:47.767 --> 00:03:52.201
The whole reason why I'm
setting this up this way is,

00:03:52.201 --> 00:03:53.901
it's interesting to now think

00:03:53.901 --> 00:03:56.066
about the probability distribution

00:03:56.066 --> 00:03:58.900
for a random variable that's defined

00:03:58.900 --> 00:04:01.533
by the number of scores
in your n attempts,

00:04:01.533 --> 00:04:04.667
or the number of scores
in your six attempts.

00:04:04.800 --> 00:04:06.963
And actually, since I've
been pushing the limit,

00:04:07.394 --> 00:04:09.533
or I've been doing longer
videos than I intend to,

00:04:09.533 --> 00:04:11.366
I will do that in the next video.


Probability of making 2 shots in 6 attempts

WEBVTT
Kind: captions
Language: en

00:00:00.376 --> 00:00:01.851
- [Voiceover] Let's say that you know

00:00:01.851 --> 00:00:04.544
your probability of making a free throw.

00:00:04.544 --> 00:00:07.142
You know that the probability...

00:00:07.144 --> 00:00:10.000
The probability, and let's
say of scoring a free throw

00:00:10.000 --> 00:00:12.589
because make and miss,
they both start with M

00:00:12.589 --> 00:00:14.249
and that can get confusing,

00:00:14.249 --> 00:00:18.241
so let's say the probability of scoring...

00:00:18.428 --> 00:00:22.199
scoring a free throw, is equal to,

00:00:22.306 --> 00:00:24.268
is going to be, say 70%.

00:00:24.268 --> 00:00:26.253
If we want to write it as a percent

00:00:26.253 --> 00:00:28.052
or we could write it as 0.7

00:00:28.052 --> 00:00:29.840
if we write it as a decimal.

00:00:29.840 --> 00:00:33.170
Let's say the probability of
missing a free throw, then

00:00:33.357 --> 00:00:34.425
and this is just going to come

00:00:34.425 --> 00:00:36.434
straight out of what we just wrote down,

00:00:36.434 --> 00:00:40.298
the probability of missing,

00:00:40.532 --> 00:00:43.005
of missing a free throw,
is just going to be

00:00:43.005 --> 00:00:44.479
100% minus this.

00:00:44.479 --> 00:00:45.860
You're either going to make or miss,

00:00:45.860 --> 00:00:47.010
you're going to score or miss,

00:00:47.010 --> 00:00:48.078
I don't want to use make in this

00:00:48.078 --> 00:00:49.343
because they both start with M

00:00:49.343 --> 00:00:52.733
so this is going to be a 30% probability,

00:00:52.733 --> 00:00:55.159
or if we write it as a decimal, 0.3

00:00:55.159 --> 00:00:56.912
One minus this is 0.7.

00:00:56.912 --> 00:00:58.619
These are the only two possibilities,

00:00:58.619 --> 00:01:00.708
so they have to add up to 100%,

00:01:00.708 --> 00:01:02.775
or they have to add up to one.

00:01:02.775 --> 00:01:06.464
Now, let's say that you were
going to take six attempts.

00:01:06.571 --> 00:01:11.271
What we are curious
about, is the probability

00:01:11.435 --> 00:01:16.435
of exactly two scores

00:01:22.127 --> 00:01:24.342
in six attempts.

00:01:29.649 --> 00:01:30.694
So let's think about what that is

00:01:30.694 --> 00:01:32.064
and I encourage you to get inspired

00:01:32.064 --> 00:01:33.643
at any point in this video

00:01:33.643 --> 00:01:35.199
you should pause it and you should try

00:01:35.199 --> 00:01:37.648
to work through what
we're asking right now,

00:01:37.648 --> 00:01:40.086
so this is what we want to figure out,

00:01:40.086 --> 00:01:42.454
the probability of exactly two scores

00:01:42.454 --> 00:01:45.308
in six attempts.

00:01:45.635 --> 00:01:47.574
So, let's think about the way,

00:01:47.574 --> 00:01:49.350
let's think about the particular ways

00:01:49.350 --> 00:01:52.032
of getting two scores in six attempts

00:01:52.032 --> 00:01:53.076
and think about the probability for

00:01:53.076 --> 00:01:54.702
any one of those particular ways,

00:01:54.702 --> 00:01:56.536
and then we can think about how many ways

00:01:56.536 --> 00:01:59.227
can we get two scores in six attempts?

00:01:59.229 --> 00:02:02.120
So, for example, you could get

00:02:02.120 --> 00:02:04.302
you could make the first two free throws,

00:02:04.302 --> 00:02:09.302
so it could be score, score and
then you miss the next four.

00:02:09.875 --> 00:02:14.875
So score, score, and then it's
miss, miss, miss, and miss.

00:02:16.631 --> 00:02:19.684
So what's the probability of
this exact thing happening?

00:02:19.684 --> 00:02:21.308
This exact thing?

00:02:21.321 --> 00:02:25.779
Well, you have a 0.7 chance of making,

00:02:25.779 --> 00:02:28.681
of scoring on the first one,

00:02:28.681 --> 00:02:33.673
then you have a 0.7 chance
of scoring on the second one,

00:02:33.673 --> 00:02:38.396
and then you have a 0.3 chance
of missing the next four.

00:02:38.665 --> 00:02:41.403
So, the probability of
this exact circumstance

00:02:41.637 --> 00:02:44.259
is going to be what I am writing down.

00:02:45.131 --> 00:02:47.708
Hopefully you don't get
the multiplication symbols

00:02:47.708 --> 00:02:49.461
confused with the decimals, I'm trying to

00:02:49.461 --> 00:02:51.398
write them a little bit higher.

00:02:51.760 --> 00:02:56.204
Times 0.3, and what is
this going to be equal to?

00:02:56.264 --> 00:02:58.573
Well, this is going to be equal to...

00:02:58.783 --> 00:03:03.783
This is going to be 0.7 squared

00:03:04.007 --> 00:03:09.007
times 0.3 to the one, two, three, fourth--

00:03:11.797 --> 00:03:13.144
to the fourth power.

00:03:13.144 --> 00:03:16.278
Now, is this the only way to
get two scores in six attempts?

00:03:16.278 --> 00:03:19.413
No, there's many ways of getting
two scores in six attempts.

00:03:19.413 --> 00:03:24.413
For example, maybe you miss the
first one, the first attempt

00:03:25.484 --> 00:03:28.839
and then you make the
second attempt, you score,

00:03:28.839 --> 00:03:31.788
then you miss the third attempt,

00:03:31.788 --> 00:03:35.280
and let's just say you
make the fourth attempt,

00:03:35.770 --> 00:03:40.019
and then you miss the next two.

00:03:40.019 --> 00:03:41.934
You miss, and you miss.

00:03:41.934 --> 00:03:45.219
This is another way to get
two scores in six attempts,

00:03:45.219 --> 00:03:48.040
and what's the probability
of this happening?

00:03:48.040 --> 00:03:49.596
Well, you'll see, it's
going to be exactly this,

00:03:49.596 --> 00:03:51.941
it's just we're multiplying
in a different order.

00:03:51.941 --> 00:03:56.941
This is going to be 0.3 times 0.7

00:03:59.765 --> 00:04:02.029
you have a 30% chance of
missing the first one,

00:04:02.029 --> 00:04:05.081
a 70% chance of making the second one,

00:04:05.187 --> 00:04:10.187
and then times 0.3, a 30%
chance of missing the third,

00:04:10.202 --> 00:04:14.321
times a 70 percent chance
of making the fourth,

00:04:14.358 --> 00:04:16.517
times a 30% chance

00:04:16.517 --> 00:04:18.862
for each of the next two misses

00:04:18.862 --> 00:04:21.263
if you wanted the exact circumstance,

00:04:23.157 --> 00:04:25.444
this is once again going to be 0.7

00:04:25.444 --> 00:04:28.103
and if you just rearrange the
order that you're multiplying,

00:04:28.103 --> 00:04:31.051
this is going to be 0.7 squared

00:04:31.051 --> 00:04:33.930
times 0.3 to the fourth power,

00:04:33.930 --> 00:04:37.216
so for any one of these particular ways

00:04:37.216 --> 00:04:39.387
to get exactly two scores in six attempts,

00:04:39.387 --> 00:04:41.325
the probability is going to be this.

00:04:41.325 --> 00:04:43.659
So, the probability of
getting exactly two scores

00:04:43.659 --> 00:04:45.470
in six attempts, well it's going to be

00:04:45.470 --> 00:04:46.898
any one of these probabilities

00:04:46.898 --> 00:04:49.069
times the number of ways

00:04:49.069 --> 00:04:54.069
you can get two scores in six attempts.

00:04:55.639 --> 00:04:56.394
Well, how...

00:04:56.394 --> 00:04:58.507
If you have out of six attempts,

00:04:58.507 --> 00:05:00.678
you're choosing two of
them to have scores,

00:05:00.678 --> 00:05:02.941
how many ways are there?

00:05:02.941 --> 00:05:03.858
Well, as you can imagine,

00:05:03.858 --> 00:05:05.484
this is a combinatorics problem,

00:05:05.484 --> 00:05:07.713
so you could write this as

00:05:07.713 --> 00:05:09.721
you could write this,
let me see how I could...

00:05:09.721 --> 00:05:12.947
You're going to take six attempts.

00:05:13.355 --> 00:05:17.534
You could write this as six choose,

00:05:17.534 --> 00:05:19.566
what we're trying to, you're
picking from six things,

00:05:19.566 --> 00:05:22.503
six attempts, and you're
picking two of them,

00:05:22.503 --> 00:05:26.624
or two of them are
going to need to be made

00:05:26.624 --> 00:05:28.969
if you want to meet these circumstances.

00:05:28.969 --> 00:05:29.991
This is going to tell us the number

00:05:29.991 --> 00:05:31.616
of different ways you can make

00:05:31.616 --> 00:05:33.496
two scores in six attempts.

00:05:33.496 --> 00:05:35.226
Of course, we can write this as kind of

00:05:35.226 --> 00:05:37.293
a binomial coefficient notation.

00:05:37.293 --> 00:05:40.982
We can write this is as six, choose two

00:05:43.538 --> 00:05:45.651
and we can just apply the formula

00:05:45.651 --> 00:05:49.343
for combinations, and if this
looks completely unfamiliar

00:05:49.343 --> 00:05:52.419
I encourage you to look up
combinations on Khan Academy

00:05:52.419 --> 00:05:53.847
and then we go into some detail

00:05:53.847 --> 00:05:56.192
on the reasoning behind the formula

00:05:56.192 --> 00:05:58.224
that makes a lot of sense.

00:05:58.224 --> 00:06:03.224
This is going to be equal to six factorial

00:06:03.355 --> 00:06:06.174
over two factorial

00:06:07.987 --> 00:06:11.259
times six minus two factorial.

00:06:11.725 --> 00:06:16.725
Six minus two factorial,

00:06:17.473 --> 00:06:20.223
I'm going to do the
factorial in green again,

00:06:20.933 --> 00:06:22.234
and what's this going to be equal to?

00:06:22.234 --> 00:06:25.477
This is going to be
equal to six times five

00:06:25.477 --> 00:06:28.822
times four times three times two,

00:06:28.822 --> 00:06:30.698
and I'll just throw in the one there

00:06:30.698 --> 00:06:32.335
although it doesn't change the value,

00:06:32.335 --> 00:06:35.456
over two times one.

00:06:37.745 --> 00:06:39.243
And six minus two is four,

00:06:39.243 --> 00:06:41.460
so that's going to be four factorial,

00:06:41.460 --> 00:06:43.097
so this right over here is four factorial

00:06:43.097 --> 00:06:48.097
so times four times three
times two times one.

00:06:48.286 --> 00:06:50.376
Well, that and that is going to cancel,

00:06:50.376 --> 00:06:55.170
and the six divided by two
is three, so this is 15.

00:06:55.170 --> 00:06:59.220
There's 15 different
ways that you could get

00:06:59.558 --> 00:07:03.308
two things out of six, I
guess, is one way to say it

00:07:08.510 --> 00:07:10.496
or there's 15 different ways

00:07:10.496 --> 00:07:12.376
that you could get two things out of six.

00:07:12.376 --> 00:07:13.456
Another way of thinking about it

00:07:13.456 --> 00:07:15.940
is there's 15 different ways

00:07:15.940 --> 00:07:19.643
to make two out of six free throws.

00:07:19.643 --> 00:07:21.594
Now, the probability for each of those

00:07:21.594 --> 00:07:23.033
is this right over here.

00:07:23.033 --> 00:07:26.516
The probability of exactly
two scores in six attempts,

00:07:26.516 --> 00:07:29.207
this is where we deserve a
little bit of a drumroll.

00:07:29.209 --> 00:07:31.193
This is going to be

00:07:33.516 --> 00:07:38.516
six choose two times 0.7 squared.

00:07:41.724 --> 00:07:45.321
This is two, you're going to make two

00:07:45.613 --> 00:07:48.304
and then it's 0.3

00:07:50.349 --> 00:07:53.832
to the fourth power.

00:07:53.832 --> 00:07:57.524
These will necessarily add up to six.

00:07:57.524 --> 00:08:00.147
So this right over here was a three,

00:08:00.147 --> 00:08:01.982
then this right over here would be a three

00:08:01.982 --> 00:08:03.874
and then this would be six minus three

00:08:03.874 --> 00:08:05.395
or three right over here.

00:08:05.395 --> 00:08:07.008
Now, what is this value?

00:08:07.008 --> 00:08:10.199
Well, it's going to be equal to

00:08:12.175 --> 00:08:15.249
we have our 15, three times five

00:08:15.309 --> 00:08:17.317
so we have this business right over here.

00:08:17.317 --> 00:08:21.100
It's going to be 15 times

00:08:22.565 --> 00:08:26.082
let's see, in yellow, 0.7 times 0.7

00:08:26.082 --> 00:08:30.271
is going to be times 0.49

00:08:30.377 --> 00:08:32.676
and let's see, three to the fourth power

00:08:32.676 --> 00:08:34.787
would be 81.

00:08:37.308 --> 00:08:39.223
But, I am multiplying four decimals,

00:08:39.223 --> 00:08:40.547
each of them have one space

00:08:40.547 --> 00:08:42.172
to the right of the decimal point

00:08:42.172 --> 00:08:44.947
so this is, I'm going to have four spaces

00:08:44.947 --> 00:08:46.073
to the right of the decimal

00:08:46.073 --> 00:08:51.073
so 0.0081

00:08:51.111 --> 00:08:54.408
so there you go, whatever this number is,

00:08:54.408 --> 00:08:56.672
and actually I might as
well get a calculator out

00:08:56.672 --> 00:08:59.595
and calculate it, so this is going to be

00:08:59.621 --> 00:09:02.151
this is going to be...

00:09:02.151 --> 00:09:03.184
Let me...

00:09:03.184 --> 00:09:08.184
So, it's 15 times 0.49

00:09:09.978 --> 00:09:14.978
times 0.0081

00:09:18.464 --> 00:09:23.464
and we get 0.059535

00:09:25.337 --> 00:09:27.531
so this is going to be equal to,

00:09:27.531 --> 00:09:29.063
let me write it down.

00:09:29.063 --> 00:09:32.186
Actually, I wish I had a
little bit more real estate

00:09:32.186 --> 00:09:35.318
right over here, but I'll
write it in a very bold color.

00:09:35.378 --> 00:09:37.073
This is going to be,

00:09:37.073 --> 00:09:39.697
well, actually, I'm kind
of out of bold colors.

00:09:39.697 --> 00:09:42.170
I'll write it in a
slightly less bold color.

00:09:42.170 --> 00:09:47.170
This is going to be equal to 0.05935

00:09:50.646 --> 00:09:52.991
if we wanted the exact
number, or we could say

00:09:52.991 --> 00:09:55.695
this is approximately, if we
round to the nearest percentage

00:09:55.695 --> 00:09:59.585
this is approximately
a six percent chance,

00:09:59.585 --> 00:10:04.275
six percent probability of
getting exactly two scores

00:10:04.275 --> 00:10:05.981
in the six attempts.

00:10:05.981 --> 00:10:07.177
I didn't say two or more,

00:10:07.177 --> 00:10:10.369
I just said exactly two
scores in the six attempts.

00:10:10.369 --> 00:10:12.424
Actually, it's a fairly low probability

00:10:12.424 --> 00:10:14.514
because I have a pretty
high free throw percentage.

00:10:14.514 --> 00:10:16.627
If someone has this high
of a free throw percentage

00:10:16.627 --> 00:10:18.101
it's actually reasonably unlikely

00:10:18.101 --> 00:10:20.063
that they're only going to make two scores

00:10:20.063 --> 00:10:23.037
in the six attempts.


Binomial distribution

WEBVTT
Kind: captions
Language: en

00:00:00.233 --> 00:00:03.135
- [Voiceover] Let's
define a random variable x

00:00:03.135 --> 00:00:07.070
as being equal to the number of heads,

00:00:07.070 --> 00:00:08.870
I'll just write capital H for short,

00:00:08.870 --> 00:00:13.709
the number of heads from flipping coin,

00:00:14.210 --> 00:00:15.951
from flipping a fair coin,

00:00:15.951 --> 00:00:17.888
we're gonna assume it's a fair coin,

00:00:17.936 --> 00:00:22.590
from flipping coin five times. Five times.

00:00:22.638 --> 00:00:25.970
Like all random variables this
is taking particular outcomes

00:00:25.970 --> 00:00:28.036
and converting them into numbers.

00:00:28.036 --> 00:00:30.033
And this random variable,
it could take on the value

00:00:30.033 --> 00:00:35.011
x equals zero, one, two,
three, four or five.

00:00:35.269 --> 00:00:37.600
And I what want to do is figure
out what's the probability

00:00:37.600 --> 00:00:40.644
that this random variable
takes on zero, can be one,

00:00:40.644 --> 00:00:43.439
can be two, can be three,
can be four, can be five.

00:00:43.627 --> 00:00:46.018
To do that, first let's
think about how many possible

00:00:46.018 --> 00:00:49.940
outcomes are there from
flipping a fair coin five times.

00:00:50.500 --> 00:00:51.939
Let's think about this.

00:00:51.939 --> 00:00:53.876
Let's write possible outcomes.

00:00:54.156 --> 00:00:58.345
Possible outcomes

00:00:59.798 --> 00:01:01.526
from five flips.

00:01:02.039 --> 00:01:04.059
From five flips.

00:01:04.059 --> 00:01:06.764
These aren't the possible
outcomes for the random variable,

00:01:06.764 --> 00:01:08.877
this is literally the
number of possible outcomes

00:01:08.877 --> 00:01:10.525
from flipping a coin five times.

00:01:10.525 --> 00:01:12.255
For example, one possible outcome could be

00:01:12.255 --> 00:01:15.921
tails, heads, tails, heads, tails.

00:01:15.981 --> 00:01:18.094
Another possible outcome could be

00:01:18.094 --> 00:01:21.459
heads, heads, heads, tails, tails.

00:01:21.577 --> 00:01:24.282
That is one of the
equally likely outcomes,

00:01:24.282 --> 00:01:26.174
that's another one of the
equally likely outcomes.

00:01:26.174 --> 00:01:28.030
How many of these are there?

00:01:28.089 --> 00:01:32.627
For each flip you have two possibilities.

00:01:33.399 --> 00:01:35.090
Let's write this down.

00:01:35.090 --> 00:01:36.469
Let me...

00:01:37.470 --> 00:01:42.470
The first flip, the first flip
there's two possibilities,

00:01:43.448 --> 00:01:46.745
times two for the second flip,
times two for the third flip.

00:01:46.745 --> 00:01:49.009
Actually maybe we'll not
use the time notation,

00:01:49.009 --> 00:01:50.750
you might get confused
with the random variable.

00:01:50.750 --> 00:01:52.863
Two possibilities for the first flip,

00:01:52.863 --> 00:01:54.651
two possibilities for the second flip,

00:01:54.651 --> 00:01:56.624
two possibilities for the third flip,

00:01:56.624 --> 00:01:58.586
two possibilities for the fourth flip,

00:01:58.586 --> 00:02:00.653
and then two possibilities
for the fifth flip,

00:02:00.653 --> 00:02:05.099
or two to the fifth equally
likely possibilities

00:02:05.099 --> 00:02:07.351
from flipping a coin five times,

00:02:07.351 --> 00:02:10.135
which is, of course, equal to 32.

00:02:10.172 --> 00:02:11.797
This is going to be helpful because for

00:02:11.797 --> 00:02:15.454
each of the values that the
random variable can take on,

00:02:15.454 --> 00:02:16.731
we just have to think about how many

00:02:16.731 --> 00:02:19.959
of these equally likely
possibilities would result

00:02:19.959 --> 00:02:22.000
in the random variable
taking on that value.

00:02:22.211 --> 00:02:24.010
Let's just delve into it to see

00:02:24.010 --> 00:02:25.599
what we're actually talking about.

00:02:25.937 --> 00:02:28.768
I'll do it in this
light, let me do it in...

00:02:28.828 --> 00:02:30.012
I'll start in blue.

00:02:30.012 --> 00:02:32.810
Let's think about the probability

00:02:32.810 --> 00:02:36.987
that our random variable
x is equal to one.

00:02:38.289 --> 00:02:39.971
Well actually, let me start with zero.

00:02:40.657 --> 00:02:44.150
The probability that our random
variable x is equal to zero.

00:02:44.291 --> 00:02:47.934
That would mean that you got
no heads out of the five flips.

00:02:48.273 --> 00:02:49.956
Well there's only one way, one out of

00:02:49.956 --> 00:02:53.495
the 32 equally likely possibilities,
that you get no heads.

00:02:53.636 --> 00:02:57.639
That's the one where
you just get five tails.

00:02:58.257 --> 00:02:59.963
So this is just going to be,

00:02:59.963 --> 00:03:01.577
this is going to be equal to

00:03:01.577 --> 00:03:05.882
one out of the 32 equally
likely possibilities.

00:03:06.476 --> 00:03:10.099
Now, for this case, to think in terms of

00:03:11.345 --> 00:03:14.590
binomial coefficients, and
combinatorics, and all of that,

00:03:14.590 --> 00:03:16.529
it's much easier to
just reason through it,

00:03:16.529 --> 00:03:19.431
but just so we can think in
terms it'll be more useful

00:03:19.431 --> 00:03:24.421
as we go into higher values
for our random variable.

00:03:25.005 --> 00:03:27.198
This is all buildup for
the binomial distribution,

00:03:27.198 --> 00:03:29.599
so you get a sense of
where the name comes from.

00:03:29.601 --> 00:03:31.422
So let's write it in those terms.

00:03:31.447 --> 00:03:34.057
This one, this one,

00:03:35.185 --> 00:03:36.740
this one right over here,

00:03:36.740 --> 00:03:39.318
one way to think about that in
combinatorics is that you had

00:03:39.318 --> 00:03:44.318
five flips and you're choosing
zero of them to be heads.

00:03:45.401 --> 00:03:48.825
Five flips and you're choosing
zero of them to be heads.

00:03:48.825 --> 00:03:52.684
Let's verify that five
choose zero is indeed one.

00:03:52.748 --> 00:03:55.835
So five choose zero. Write it over here.

00:03:55.837 --> 00:04:00.837
Five choose zero is equal
to five factorial over,

00:04:02.303 --> 00:04:07.049
over five minus zero factorial.

00:04:08.103 --> 00:04:10.812
Well actually over zero factorial

00:04:10.812 --> 00:04:14.119
times five minus zero factorial.

00:04:16.419 --> 00:04:19.229
Well zero factorial is one, by definition,

00:04:19.229 --> 00:04:24.184
so this is going to be five
factorial, over five factorial,

00:04:24.418 --> 00:04:27.028
which is going to be equal to one.

00:04:27.065 --> 00:04:28.562
Once again I like reasoning through it

00:04:28.562 --> 00:04:30.547
instead of blindly applying a formula,

00:04:30.547 --> 00:04:32.022
but I just wanted to show you that

00:04:32.022 --> 00:04:34.237
these two ideas are consistent.

00:04:34.645 --> 00:04:35.539
Let's keep going.

00:04:35.539 --> 00:04:36.897
I'm going to do x equals one

00:04:36.897 --> 00:04:38.523
all the way up to x equals five.

00:04:38.523 --> 00:04:40.798
If you are inspired, and I
encourage you to be inspired,

00:04:40.798 --> 00:04:43.085
try to fill out the whole thing,

00:04:43.085 --> 00:04:44.304
what's the probability that x equals

00:04:44.304 --> 00:04:45.904
one, two, three, four or five.

00:04:46.127 --> 00:04:48.936
So let's go to the
probability that x equals two.

00:04:48.936 --> 00:04:50.559
Or sorry, that x equals one.

00:04:51.002 --> 00:04:56.002
The probability that x equals
one is going to be equal to...

00:04:57.202 --> 00:04:58.955
Well how do you get one head?

00:04:58.955 --> 00:05:01.079
It could be, the first one could be head

00:05:01.079 --> 00:05:02.716
and then the rest of
them are gonna be tails.

00:05:02.716 --> 00:05:04.538
The second one could be head

00:05:04.538 --> 00:05:06.199
and then the rest of
them are gonna be tails.

00:05:06.199 --> 00:05:07.859
I could write them all
out but you can see that

00:05:07.859 --> 00:05:10.981
there's five different
places to have that one head.

00:05:10.981 --> 00:05:13.884
So five out of the 32
equally likely outcomes

00:05:13.884 --> 00:05:16.053
involve one head.

00:05:16.356 --> 00:05:17.599
Let me write that down.

00:05:17.599 --> 00:05:21.132
This is going to be equal to

00:05:21.132 --> 00:05:25.224
five out of 32 equally likely outcomes.

00:05:26.166 --> 00:05:28.291
Which of course is the same
thing, this is going to be

00:05:28.291 --> 00:05:30.566
the same thing as saying I got five flips,

00:05:30.566 --> 00:05:35.208
and I'm choosing one of them to be heads.

00:05:35.593 --> 00:05:37.692
So that over 32.

00:05:39.225 --> 00:05:41.803
You could verify that five factorial

00:05:41.803 --> 00:05:46.168
over one factorial times five minus--

00:05:46.168 --> 00:05:47.852
Actually let me just do it just so that

00:05:47.852 --> 00:05:49.812
you don't have to take my word for it.

00:05:49.965 --> 00:05:54.965
So five choose one is
equal to five factorial

00:05:56.001 --> 00:05:58.971
over one factorial, which is just one,

00:05:59.101 --> 00:06:03.940
times five minus four-- Sorry,
five minus one factorial.

00:06:04.990 --> 00:06:09.003
Which is equal to five
factorial over four factorial,

00:06:09.003 --> 00:06:12.345
which is just going to be equal to five.

00:06:12.683 --> 00:06:14.690
All right, we're making good progress.

00:06:15.052 --> 00:06:19.869
Now in purple let's think
about the probability that

00:06:19.869 --> 00:06:22.712
our random variable x is equal to two.

00:06:22.946 --> 00:06:25.105
Well this is going to be equal to,

00:06:25.105 --> 00:06:27.866
and now I'll actually
resort to the combinatorics.

00:06:29.460 --> 00:06:30.840
You have five flips

00:06:30.840 --> 00:06:33.601
and you're choosing two
of them to be heads.

00:06:34.241 --> 00:06:37.457
Over 32 equally likely possibilities.

00:06:37.457 --> 00:06:38.734
This is the number of possibilities

00:06:38.734 --> 00:06:41.137
that result in two heads.

00:06:41.137 --> 00:06:43.912
Two of the five flips
have chosen to be heads,

00:06:43.912 --> 00:06:47.893
I guess you can think of it
that way, by the random gods,

00:06:47.893 --> 00:06:49.716
or whatever you want to say.

00:06:49.716 --> 00:06:53.315
This is the fraction of the 32
equally likely possibilities,

00:06:53.315 --> 00:06:55.300
so this is the probability
that x equals two.

00:06:55.300 --> 00:06:58.595
What's this going to be?
I'll do it right over here.

00:06:58.759 --> 00:06:59.677
And actually no reason for me

00:06:59.677 --> 00:07:01.093
to have to keep switching colors.

00:07:01.093 --> 00:07:06.093
So five choose two is going
to be equal to five factorial

00:07:07.455 --> 00:07:10.518
over two factorial

00:07:10.949 --> 00:07:14.418
times five minus two factorial.

00:07:15.151 --> 00:07:16.719
Five minus two factorial.

00:07:16.719 --> 00:07:18.553
So this is five factorial

00:07:18.553 --> 00:07:22.568
over two factorial times three factorial.

00:07:22.895 --> 00:07:24.253
And this is going to be equal to

00:07:24.253 --> 00:07:27.991
five times four times three times two,

00:07:27.991 --> 00:07:29.012
I could write times one

00:07:29.012 --> 00:07:31.251
but that doesn't really
do anything for us.

00:07:31.369 --> 00:07:33.819
Then two factorial's just going to be two.

00:07:33.819 --> 00:07:37.801
Then the three factorial
is three times two.

00:07:37.801 --> 00:07:38.996
I could write times one,

00:07:38.996 --> 00:07:40.898
but once again doesn't do anything for us.

00:07:40.923 --> 00:07:42.514
That cancels with that.

00:07:42.514 --> 00:07:44.799
Four divided by two is two.

00:07:44.801 --> 00:07:46.693
Five times two is 10.

00:07:46.693 --> 00:07:48.235
So this is equal to 10.

00:07:48.713 --> 00:07:51.300
This right over here is equal to 10/32.

00:07:51.522 --> 00:07:53.113
10/32.

00:07:53.113 --> 00:07:54.680
And obviously we could
simplify this fraction,

00:07:54.680 --> 00:07:56.201
but I like to leave it this way

00:07:56.201 --> 00:07:58.534
because we're now thinking
everything is in terms of 32nds.

00:07:58.534 --> 00:08:00.740
There's a 1/32 chance x equals zero,

00:08:00.740 --> 00:08:03.097
5/32 chance that x equals one

00:08:03.097 --> 00:08:05.742
and a 10/32 chance that x equals two.

00:08:05.999 --> 00:08:07.959
Let's keep on going.

00:08:08.297 --> 00:08:10.153
I'll go in orange.

00:08:10.561 --> 00:08:13.220
What is the probability that

00:08:13.220 --> 00:08:16.016
our random variable x is equal to three?

00:08:16.400 --> 00:08:18.339
Well this is going to be five,

00:08:18.339 --> 00:08:21.299
out of the five flips we're
going to need to choose

00:08:21.299 --> 00:08:22.994
three of them to be heads to figure out

00:08:22.994 --> 00:08:25.674
which of the possibilities
involve exactly three heads.

00:08:25.931 --> 00:08:28.936
And this is over 32 equally
likely possibilities.

00:08:29.136 --> 00:08:31.397
And this is going to be equal to,

00:08:31.829 --> 00:08:36.829
five choose three is
equal to five factorial

00:08:37.192 --> 00:08:42.031
over three factorial times
five minus three factorial.

00:08:42.614 --> 00:08:43.670
Let me just write it down.

00:08:43.670 --> 00:08:45.493
Five minus three factorial,

00:08:45.493 --> 00:08:47.222
which is equal to five factorial

00:08:47.222 --> 00:08:51.644
over three factorial times two factorial.

00:08:53.501 --> 00:08:55.209
That's exactly what we had up here

00:08:55.209 --> 00:08:56.974
and we just swapped three and the two,

00:08:56.974 --> 00:08:59.618
so this also is going to be equal to 10.

00:08:59.794 --> 00:09:03.577
So this is also going
to be equal to 10/32.

00:09:03.938 --> 00:09:05.761
All right, two more to go.

00:09:05.761 --> 00:09:06.841
And I think you're going to start seeing

00:09:06.841 --> 00:09:08.048
a little bit of a symmetry here.

00:09:08.048 --> 00:09:11.006
One, five, 10, 10, let's keep going.

00:09:11.565 --> 00:09:14.421
Let's keep going, and I
haven't used white yet.

00:09:14.421 --> 00:09:15.501
Maybe I'll use white.

00:09:15.501 --> 00:09:20.501
The probability that our random
variable x is equal to four.

00:09:22.048 --> 00:09:24.463
Well, out of our five
flips we want to select

00:09:24.463 --> 00:09:27.394
four of them to be heads,
or out of the five--

00:09:30.778 --> 00:09:32.798
We're obviously not actively selecting.

00:09:32.798 --> 00:09:34.621
One way to think of it,
we want to figure out

00:09:34.621 --> 00:09:36.641
the possibilities that
involve out of the five flips,

00:09:36.641 --> 00:09:38.916
four of them are chosen to be heads,

00:09:38.916 --> 00:09:40.563
or four of them are heads.

00:09:40.704 --> 00:09:43.581
And this is over 32 equally
likely possibilities.

00:09:43.757 --> 00:09:48.757
So five choose four is
equal to five factorial

00:09:49.922 --> 00:09:54.842
over four factorial times
five minus four factorial

00:09:55.064 --> 00:09:56.678
which is equal to,

00:09:56.678 --> 00:09:58.117
well that's just going
to be five factorial,

00:09:58.117 --> 00:10:00.269
this is going to be one
factorial right over here.

00:10:01.407 --> 00:10:03.005
That doesn't change the value,

00:10:03.005 --> 00:10:05.292
you just multiply one
factorial times four factorial,

00:10:05.292 --> 00:10:07.765
so it's five factorial
over four factorial,

00:10:07.765 --> 00:10:09.806
which is equal to five.

00:10:10.110 --> 00:10:13.405
So once again this is 5/32.

00:10:13.407 --> 00:10:14.707
And you could have reasoned through this

00:10:14.707 --> 00:10:16.471
because if you're saying
you want five heads,

00:10:16.471 --> 00:10:18.530
that means you have one tail.

00:10:19.147 --> 00:10:22.206
There's five different places
you could put that one tail.

00:10:22.206 --> 00:10:24.479
There are five
possibilities with one tail.

00:10:24.479 --> 00:10:26.790
Five of the 32 equally likely.

00:10:26.850 --> 00:10:28.417
And then, and you could probably guess

00:10:28.417 --> 00:10:30.286
what we're gonna get for x equals five

00:10:30.286 --> 00:10:31.993
because having five heads means

00:10:31.993 --> 00:10:33.815
you have zero tails,

00:10:33.815 --> 00:10:36.207
and there's only gonna be
one possibility out of the 32

00:10:36.207 --> 00:10:39.409
with zero tails, where you have all heads.

00:10:39.666 --> 00:10:41.255
Let's write that down.

00:10:41.477 --> 00:10:43.286
The probability,

00:10:43.369 --> 00:10:48.369
the probability that our random
variable x is equal to five.

00:10:48.768 --> 00:10:50.114
So we have all five heads.

00:10:50.114 --> 00:10:51.530
You could say this is five

00:10:51.530 --> 00:10:55.638
and we're choosing five
of them to be heads.

00:10:56.174 --> 00:10:58.471
Out of the 32 equally
likely possibilities.

00:10:58.682 --> 00:11:02.197
Well five choose five,
that's going to be...

00:11:02.675 --> 00:11:04.161
Let me just write it here since

00:11:04.161 --> 00:11:05.775
I've done it for all of the other ones.

00:11:05.775 --> 00:11:10.775
Five choose five is five
factorial over five factorial

00:11:10.976 --> 00:11:13.737
times five minus five factorial.

00:11:13.971 --> 00:11:15.898
Well this right over
here is zero factorial,

00:11:15.898 --> 00:11:17.453
which is equal to one,

00:11:17.453 --> 00:11:19.113
so this whole thing simplifies to one.

00:11:19.113 --> 00:11:22.190
This is going to be one out of-- 1/32.

00:11:22.190 --> 00:11:24.953
So you see the symmetry. 1/32, 1/32.

00:11:24.953 --> 00:11:28.677
5/32, 5/32; 10/32, 10/32.

00:11:28.679 --> 00:11:31.454
And that makes sense because
the probability of getting five

00:11:31.454 --> 00:11:35.041
heads is the same as the
probability of getting zero tails,

00:11:35.041 --> 00:11:36.620
and the probability of getting zero tails

00:11:36.620 --> 00:11:40.971
should be the same as the
probability of getting zero heads.

00:11:41.298 --> 00:11:43.179
I'll leave you there for this video.

00:11:43.179 --> 00:11:46.197
In the next video we'll graphically
represent this and we'll

00:11:46.197 --> 00:11:49.875
see the probability distribution
for this random variable.


Visualizing a binomial distribution

WEBVTT
Kind: captions
Language: en

00:00:00.409 --> 00:00:01.123
- [Voiceover] In the last video,

00:00:01.123 --> 00:00:02.874
we set up a random variable x,

00:00:02.874 --> 00:00:04.409
which was defined as the number of heads

00:00:04.409 --> 00:00:07.291
from flipping a fair coin five times,

00:00:07.291 --> 00:00:08.824
and then we figured out the probability

00:00:08.824 --> 00:00:10.602
that our random variable
could take on the value

00:00:10.602 --> 00:00:13.767
zero, one, two, three, four, or five,

00:00:13.767 --> 00:00:15.942
and just to visualize that, in this video,

00:00:15.942 --> 00:00:17.257
we will actually plot these,

00:00:17.257 --> 00:00:18.374
and we'll get a sense of

00:00:18.374 --> 00:00:21.675
this random variable's
probability distribution.

00:00:21.675 --> 00:00:23.700
So let's do that.

00:00:24.454 --> 00:00:25.875
Actually, maybe I'll do it like this,

00:00:25.875 --> 00:00:30.243
so that we can see the probabilities.

00:00:30.243 --> 00:00:34.373
Actually, let me erase this
business right over here.

00:00:34.373 --> 00:00:36.542
Whoops, that's not working.

00:00:37.742 --> 00:00:39.324
Here, that might work.

00:00:39.324 --> 00:00:42.706
Let me erase this real fast,

00:00:42.706 --> 00:00:44.824
these little scribbles
that I had off-screen,

00:00:44.824 --> 00:00:48.874
and then we can actually
plot the distribution.

00:00:48.874 --> 00:00:51.708
Alright, so at one axis,

00:00:51.708 --> 00:00:54.025
I'm gonna put all of
the different outcomes.

00:00:54.025 --> 00:00:56.844
So let me...

00:00:56.844 --> 00:01:00.833
That looks like a pretty straight line,

00:01:01.187 --> 00:01:06.148
and then at this axis, I'm
going to plot the probabilities.

00:01:06.825 --> 00:01:10.791
And that looks like a
pretty straight line,

00:01:10.791 --> 00:01:13.041
and let's see what the probabilities are.

00:01:13.041 --> 00:01:15.925
We have, let's see,

00:01:15.925 --> 00:01:17.919
it's all gonna be in terms of 32nds,

00:01:17.919 --> 00:01:20.358
and we get as high as 10/32,

00:01:20.358 --> 00:01:23.235
so let's say this right over here is

00:01:24.081 --> 00:01:26.175
10/32,

00:01:27.775 --> 00:01:31.673
half-way up there, we have two 5/32,

00:01:31.673 --> 00:01:34.424
so let's see, that looks like about half,

00:01:34.424 --> 00:01:38.641
that right over there is 5/32,

00:01:38.641 --> 00:01:41.720
and then 1/32 would be about this,

00:01:41.720 --> 00:01:43.742
that's one, two, let's see...

00:01:43.742 --> 00:01:47.391
If I were to split it
up, one, two, three...

00:01:47.391 --> 00:01:49.664
Actually let me do it a little bit...

00:01:49.664 --> 00:01:53.240
One, two, three,

00:01:53.240 --> 00:01:55.154
four, five.

00:01:55.154 --> 00:01:59.241
Alright, so let's say this
is 1/32 right over here,

00:01:59.241 --> 00:02:02.943
and our probabilities,
so this right over here,

00:02:02.943 --> 00:02:06.083
probability, so this is the values

00:02:06.083 --> 00:02:08.274
that the random variable could take on,

00:02:08.274 --> 00:02:11.193
so I'll just make a little histogram here,

00:02:11.193 --> 00:02:14.914
so x equals zero,

00:02:14.914 --> 00:02:17.482
and the probability there,

00:02:17.482 --> 00:02:19.775
and actually since I
want to do a histogram,

00:02:19.775 --> 00:02:22.224
it will look like this,

00:02:22.224 --> 00:02:24.541
so let me do it a little bit different.

00:02:24.541 --> 00:02:27.078
So, put it right here,

00:02:27.078 --> 00:02:30.479
so x is equal to zero.

00:02:30.479 --> 00:02:33.958
Right there, the probability is 1/32,

00:02:33.958 --> 00:02:36.091
and I can shade that in.

00:02:36.091 --> 00:02:38.641
Now, over the probability
that x equals one,

00:02:38.641 --> 00:02:41.492
x equals one is 5/32,

00:02:41.492 --> 00:02:43.942
so let me draw that,

00:02:43.942 --> 00:02:46.712
so 5/32,

00:02:47.436 --> 00:02:50.153
so, put the bar there,

00:02:50.799 --> 00:02:52.918
so let me shade that in,

00:02:52.918 --> 00:02:55.169
so this right over here

00:02:55.169 --> 00:02:58.368
is the probability that x is equal to one,

00:02:58.368 --> 00:03:01.184
that we get one, that one, exactly one,

00:03:01.184 --> 00:03:04.768
out of the five flips result in heads.

00:03:04.768 --> 00:03:07.585
Now we have the probability x equals two.

00:03:07.585 --> 00:03:10.286
x equals two is 10/32,

00:03:10.286 --> 00:03:12.846
so that's going to look like this.

00:03:17.847 --> 00:03:22.122
Alright, my best attempt
at hand-drawing it.

00:03:24.615 --> 00:03:28.949
Somehow I like the aesthetics
of hand-drawn things more.

00:03:28.949 --> 00:03:30.716
Sometimes if you just get
a computer to graph it,

00:03:30.716 --> 00:03:31.531
I don't know, sometimes,

00:03:31.531 --> 00:03:33.633
it loses a little bit of its personality.

00:03:33.633 --> 00:03:37.449
Alright, so that right over
there is the probability

00:03:37.449 --> 00:03:41.666
that our random variable
x is equal to two.

00:03:41.666 --> 00:03:43.766
Then we have the probability
that x equals three,

00:03:43.766 --> 00:03:45.951
which is also 10/32.

00:03:45.951 --> 00:03:50.546
So that is also 10/32,

00:03:50.546 --> 00:03:52.670
so let me draw that.

00:03:53.747 --> 00:03:57.394
So this is also 10/32,

00:03:58.025 --> 00:03:59.904
shade this in.

00:04:00.889 --> 00:04:04.383
Dum-de-dum-de-dum, alright.

00:04:04.383 --> 00:04:08.567
I find this strangely
therapeutic. (chuckling)

00:04:08.567 --> 00:04:11.193
Alright, so this is the probability

00:04:11.193 --> 00:04:14.093
that x is equal to three.

00:04:14.093 --> 00:04:17.757
Now x equals four, that's 5/32.

00:04:17.757 --> 00:04:20.854
So we go back right over here,

00:04:20.854 --> 00:04:23.193
and that's 5/32.

00:04:25.117 --> 00:04:27.697
So, shade that one in.

00:04:30.236 --> 00:04:33.350
So this right over here
is x is equal to four,

00:04:33.350 --> 00:04:34.462
and then finally, the probability

00:04:34.462 --> 00:04:38.116
that x equals five is 1/32 again.

00:04:38.501 --> 00:04:41.351
Same level as this right over here,

00:04:41.351 --> 00:04:44.936
shade it in, so this right over here is,

00:04:44.936 --> 00:04:47.334
our random variable x equaling five.

00:04:47.334 --> 00:04:48.735
And so, when you visually show

00:04:48.735 --> 00:04:50.384
this probability distribution,
it's important to realize,

00:04:50.384 --> 00:04:52.701
this is a discrete
probability distribution.

00:04:52.701 --> 00:04:54.951
This is a discrete random variable.

00:04:54.951 --> 00:04:56.810
It can only take on

00:04:56.810 --> 00:05:00.238
a finite number of values.

00:05:00.238 --> 00:05:01.084
Actually, I should say it's a

00:05:01.084 --> 00:05:02.834
finite discrete random variable.

00:05:02.834 --> 00:05:04.784
You could have something that
takes on discrete values,

00:05:04.784 --> 00:05:06.485
but in theory, it could take on

00:05:06.485 --> 00:05:08.151
an infinite number of discrete values.

00:05:08.151 --> 00:05:10.301
You could just keep counting
higher and higher and higher.

00:05:10.301 --> 00:05:11.518
But this is discrete, in that

00:05:11.518 --> 00:05:14.035
it's these whole, these particular values.

00:05:14.035 --> 00:05:15.818
It can't take on any value in between,

00:05:15.818 --> 00:05:16.876
and it's also finite.

00:05:16.876 --> 00:05:19.262
It can only take on x
equals zero, x equals one,

00:05:19.262 --> 00:05:22.035
x equals two, x equals
three, x equals four,

00:05:22.035 --> 00:05:23.834
or x equals five,

00:05:23.834 --> 00:05:28.284
and you see when you plot
its probability distribution,

00:05:28.284 --> 00:05:30.955
this discrete probability distribution,

00:05:30.955 --> 00:05:35.578
it starts at 1/32, it goes up,
and then it comes back down,

00:05:35.578 --> 00:05:37.568
and it has this symmetry,

00:05:37.568 --> 00:05:40.377
and a distribution like
this, this right over here,

00:05:40.377 --> 00:05:41.985
a discrete distribution like this,

00:05:41.985 --> 00:05:44.584
we call this a binomial distribution,

00:05:44.584 --> 00:05:45.700
and we'll talk in the future

00:05:45.700 --> 00:05:50.235
about why it's called a
binomial distribution,

00:05:50.235 --> 00:05:51.851
but a big clue...

00:05:51.851 --> 00:05:52.550
Actually, I'll tell you why

00:05:52.550 --> 00:05:54.351
it's called a binomial distribution,

00:05:54.351 --> 00:05:56.318
is that these probabilities,

00:05:56.318 --> 00:05:59.366
you can get them using
binomial coefficients,

00:05:59.366 --> 00:06:00.784
using combinatorics.

00:06:00.784 --> 00:06:02.117
In another video, we'll talk about,

00:06:02.117 --> 00:06:04.235
especially when we talk
about the binomial theorem,

00:06:04.235 --> 00:06:07.554
why we even call those
things binomial coefficients.

00:06:07.554 --> 00:06:11.667
It's really based on taking
powers of binomials in algebra,

00:06:11.667 --> 00:06:15.186
but this is a very, very, very,
very important distribution.

00:06:15.186 --> 00:06:16.701
It's very important in statistics,

00:06:16.701 --> 00:06:21.337
because for a lot of discrete processes,

00:06:21.337 --> 00:06:24.568
one might assume that the
underlying distribution

00:06:24.568 --> 00:06:26.485
is a binomial distribution,
and when we get

00:06:26.485 --> 00:06:28.605
further into statistics,

00:06:28.605 --> 00:06:31.695
we'll talk why people do that.

00:06:31.695 --> 00:06:34.729
Now, if you were to have

00:06:34.729 --> 00:06:36.735
much more than five cases here,

00:06:36.735 --> 00:06:38.217
if, instead of saying
that the number of heads

00:06:38.217 --> 00:06:40.018
from flipping a coin five times, you said,

00:06:40.018 --> 00:06:41.568
x is equal to the number of heads

00:06:41.568 --> 00:06:45.200
of flipping a coin five million times,

00:06:45.200 --> 00:06:47.926
then, you can imagine,
you'd have much, much...

00:06:47.926 --> 00:06:50.018
The bars would get narrower and narrower

00:06:50.018 --> 00:06:53.117
relative to the whole hump,
and what it would start to do,

00:06:53.117 --> 00:06:57.468
it would start to approach
something that looks really,

00:06:57.468 --> 00:07:00.082
something that looks
really like a bell curve.

00:07:00.082 --> 00:07:00.935
I think I'm gonna do that in a color

00:07:00.935 --> 00:07:03.867
that you can see better,
that I haven't used yet.

00:07:03.867 --> 00:07:05.664
So, it would start to look...

00:07:05.664 --> 00:07:07.993
So if you had more and more of these,

00:07:07.993 --> 00:07:10.865
if you had more and more
of these possibilities,

00:07:10.865 --> 00:07:13.735
it's going to start approaching

00:07:13.735 --> 00:07:15.935
what looks like a bell curve,

00:07:15.935 --> 00:07:18.485
and you've probably heard
the notion of a bell curve,

00:07:18.485 --> 00:07:21.384
and the bell curve is
a normal distribution.

00:07:21.384 --> 00:07:24.402
So one way to think about it,
is the normal distribution

00:07:24.402 --> 00:07:26.034
is a probability density function.

00:07:26.034 --> 00:07:27.367
It's a continuous case.

00:07:27.367 --> 00:07:29.318
So, the yellow one,

00:07:29.318 --> 00:07:32.500
that we're approaching
a normal distribution,

00:07:32.500 --> 00:07:35.450
and a normal distribution, in
kind of the classical sense,

00:07:35.450 --> 00:07:39.151
is going to keep going on and on,

00:07:39.151 --> 00:07:42.818
normal distribution, and
it's related to the binomial.

00:07:42.818 --> 00:07:45.051
You know, a lot of times in statistics,

00:07:45.051 --> 00:07:47.019
people will assume a normal distribution,

00:07:47.019 --> 00:07:48.735
because you can say, okay,
it's the product of kind of

00:07:48.735 --> 00:07:52.885
an almost an infinite number
of random processes happening.

00:07:52.885 --> 00:07:55.318
Here, we're taking a coin, and
we're flipping it five times,

00:07:55.318 --> 00:07:57.801
but if you imagine molecules interacting,

00:07:57.801 --> 00:07:58.986
or humans interacting, you're saying,

00:07:58.986 --> 00:08:01.652
oh, there's almost an infinite
number of interactions,

00:08:01.652 --> 00:08:04.200
and then that's going to result
in a normal distribution,

00:08:04.200 --> 00:08:07.518
which is very, very important
in science and statistics.

00:08:07.518 --> 00:08:12.485
Binomial distribution is the
discrete version of that,

00:08:12.485 --> 00:08:14.068
and one little point of notion,

00:08:14.068 --> 00:08:16.134
these are where the distributions are,

00:08:16.134 --> 00:08:17.084
this is where they come from,

00:08:17.084 --> 00:08:17.901
this is how they're related.

00:08:17.901 --> 00:08:21.377
If you kind of think about as
you get more and more trials,

00:08:21.377 --> 00:08:23.217
the binomial distribution
is going to really

00:08:23.217 --> 00:08:25.001
approach the normal distribution,

00:08:25.001 --> 00:08:26.000
but it's really important to think about

00:08:26.000 --> 00:08:27.269
where these things come from,

00:08:27.269 --> 00:08:29.385
and we'll talk about it
much more in a statistics,

00:08:29.385 --> 00:08:31.569
because it is reasonable to assume

00:08:31.569 --> 00:08:33.268
an underlying binomial distribution,

00:08:33.268 --> 00:08:34.300
or normal distribution,

00:08:34.300 --> 00:08:36.569
for a lot of different types of processes,

00:08:36.569 --> 00:08:39.401
but sometimes it's not, and
even in things like economics,

00:08:39.401 --> 00:08:41.117
sometimes people assume
a normal distribution

00:08:41.117 --> 00:08:42.818
when it's actually much more likely

00:08:42.818 --> 00:08:44.584
that the things on the
ends are going to happen,

00:08:44.584 --> 00:08:46.685
which might lead to things
like economic crises,

00:08:46.685 --> 00:08:48.050
or whatever else.

00:08:48.050 --> 00:08:49.652
But anyway, I don't want to get off-topic.

00:08:49.652 --> 00:08:51.750
The whole point here
is just to appreciate,

00:08:51.750 --> 00:08:53.519
hey, we started with this random variable,

00:08:53.519 --> 00:08:56.018
the number of heads from
flipping a coin five times,

00:08:56.018 --> 00:08:57.884
and we plotted it, and
we were able to see,

00:08:57.884 --> 00:09:00.341
we were able to visualize
this binomial distribution,

00:09:00.341 --> 00:09:01.351
and I'm kind of telling you,

00:09:01.351 --> 00:09:02.652
I haven't really shown you,

00:09:02.652 --> 00:09:05.401
that if you were to have
many, many more flips,

00:09:05.401 --> 00:09:08.050
and you defined the random
variable in a similar way,

00:09:08.050 --> 00:09:11.551
then this histogram, this bar chart,

00:09:11.551 --> 00:09:14.318
is gonna look a lot
more like a bell curve,

00:09:14.318 --> 00:09:16.318
and if you had essentially
an infinite number of them,

00:09:16.318 --> 00:09:16.902
you would start having

00:09:16.902 --> 00:09:18.694
a continuous probability distribution,

00:09:18.694 --> 00:09:20.784
or, I should say,
probability density function,

00:09:20.784 --> 00:09:24.784
and that would get us closer
to a normal distribution.


Polynomial remainder theorem to test factor

WEBVTT
Kind: captions
Language: en

00:00:00.272 --> 00:00:01.200
- [Voiceover] So we're asked,

00:00:01.200 --> 00:00:03.250
Is the expression x minus three,

00:00:03.250 --> 00:00:06.362
is this a factor of this
fourth degree polynomial?

00:00:06.362 --> 00:00:09.055
And you could solve this by
doing algebraic long division

00:00:09.055 --> 00:00:10.354
by taking all of this business

00:00:10.370 --> 00:00:12.422
and dividing it by x minus three

00:00:12.422 --> 00:00:14.371
and figuring out if you have a remainder.

00:00:14.371 --> 00:00:16.091
If you do end up with a remainder

00:00:16.091 --> 00:00:19.093
then this is not a factor of this.

00:00:19.093 --> 00:00:21.083
But if you don't have a
remainder then that means

00:00:21.083 --> 00:00:23.444
that this divides fully
into this right over here

00:00:23.444 --> 00:00:25.772
without a remainder which
means it is a factor.

00:00:25.772 --> 00:00:28.234
So if the remainder is equal to zero,

00:00:28.234 --> 00:00:29.812
the remainder is equal to zero,

00:00:29.812 --> 00:00:31.160
if and only if,

00:00:31.160 --> 00:00:32.437
it's a factor.

00:00:32.437 --> 00:00:34.689
It is a factor.

00:00:34.689 --> 00:00:36.407
And we know a very fast way

00:00:36.407 --> 00:00:38.497
of calculating the remainder

00:00:38.497 --> 00:00:40.486
of when you take some polynomial

00:00:40.486 --> 00:00:44.325
and you divide it by a first
degree expression like this.

00:00:44.325 --> 00:00:45.369
I guess you could say when you divide it

00:00:45.369 --> 00:00:47.946
by a first degree polynomial like this.

00:00:47.946 --> 00:00:49.967
The polynomial remainder theorem,

00:00:49.967 --> 00:00:52.359
the polynomial remainder theorem tells us

00:00:52.359 --> 00:00:57.359
that if we take some polynomial, p of x

00:00:57.653 --> 00:01:02.133
and we were to divide it by some x minus a

00:01:02.133 --> 00:01:06.358
then the remainder is
just going to be equal to

00:01:06.358 --> 00:01:10.144
our polynomial evaluated at

00:01:10.144 --> 00:01:12.247
our polynomial evaluated at a.

00:01:12.247 --> 00:01:14.671
So let's just see what's a in this case.

00:01:14.671 --> 00:01:17.761
Well in this case our a is positive three.

00:01:17.761 --> 00:01:21.778
So let's just evaluate our
polynomial at x equals 3,

00:01:21.778 --> 00:01:23.844
if what we get is equal to zero

00:01:23.844 --> 00:01:25.237
that means our remainder is zero

00:01:25.237 --> 00:01:27.745
and that means that x
minus three is a factor.

00:01:27.745 --> 00:01:28.952
If we get some other remainder

00:01:28.952 --> 00:01:30.531
that means well we have
a non-zero remainder

00:01:30.531 --> 00:01:33.549
and this isn't a factor,
so let's try it out.

00:01:33.549 --> 00:01:36.568
So, we're gonna have,

00:01:36.568 --> 00:01:39.122
so I'm just gonna do it all in magenta.

00:01:39.122 --> 00:01:40.701
It might be a little
computationally intensive.

00:01:40.701 --> 00:01:44.415
So it's going to be two times
three to the fourth power,

00:01:44.415 --> 00:01:46.734
three to the fourth, three
to (mumbles), that's 81.

00:01:46.734 --> 00:01:48.572
81.

00:01:48.572 --> 00:01:50.523
Minus 11

00:01:50.523 --> 00:01:53.209
Yeah, this is gonna get a
little computationally intensive

00:01:53.209 --> 00:01:55.492
but let's see if we can power through it.

00:01:55.492 --> 00:01:58.348
11 times 27,

00:01:58.348 --> 00:01:59.925
I probably should have
picked a simpler example,

00:01:59.925 --> 00:02:01.250
but let's just keep going.

00:02:01.250 --> 00:02:04.826
Plus 15 times nine.

00:02:04.826 --> 00:02:07.728
Plus four times three is 12.

00:02:07.728 --> 00:02:08.799
Minus 12

00:02:08.799 --> 00:02:12.929
So lucky for us, at least those
last two terms cancel out.

00:02:12.929 --> 00:02:14.067
And so this is going to be

00:02:14.067 --> 00:02:16.273
the rest from here is arithmetic.

00:02:16.273 --> 00:02:20.260
Two times 81 is 162.

00:02:20.260 --> 00:02:23.615
Now let's think about what 27 times 11 is.

00:02:23.615 --> 00:02:27.832
So let's see, 27 times
10 is going to be 270.

00:02:27.832 --> 00:02:32.828
270 plus another 27 is minus 297.

00:02:32.828 --> 00:02:36.171
297, did I do that, yeah, 270

00:02:36.171 --> 00:02:40.260
So 27 times 10 is 270 plus 27, 297

00:02:40.260 --> 00:02:41.581
Yep, that's right.

00:02:41.581 --> 00:02:44.214
And then we have,

00:02:44.214 --> 00:02:46.357
I'm prone to make careless errors here,

00:02:46.357 --> 00:02:49.871
see 90 plus 45 is 135.

00:02:49.871 --> 00:02:53.516
So plus 135.

00:02:53.516 --> 00:02:54.515
And let's see,

00:02:54.515 --> 00:02:56.094
if I were to take

00:02:56.094 --> 00:02:59.413
if I were to take 162 and 135,

00:02:59.413 --> 00:03:03.193
that's going to give me 297

00:03:03.193 --> 00:03:05.713
minus 297.

00:03:05.713 --> 00:03:08.864
Minus 200, we do it in that green color,

00:03:08.864 --> 00:03:11.684
minus 297.

00:03:11.684 --> 00:03:14.244
And we do indeed equal zero.

00:03:14.244 --> 00:03:15.954
So the remainder,

00:03:15.954 --> 00:03:18.030
if I were to divide this by this,

00:03:18.030 --> 00:03:19.244
is equal to zero.

00:03:19.244 --> 00:03:21.704
So x minus three is indeed a factor

00:03:21.704 --> 00:03:25.055
of all of this business.


Constructing a polynomial that has a certain factor

WEBVTT
Kind: captions
Language: en

00:00:00.694 --> 00:00:04.791
- So for what value of c, or values of c,

00:00:04.791 --> 00:00:07.987
is x minus five?

00:00:08.655 --> 00:00:10.198
Let me write it this way.

00:00:10.198 --> 00:00:15.198
For what c is x minus five a factor

00:00:22.133 --> 00:00:24.878
of p of x?

00:00:24.878 --> 00:00:26.480
I encourage you to pause the video

00:00:26.480 --> 00:00:27.943
and try to work through it.

00:00:27.943 --> 00:00:29.568
That p is actually a lower case p.

00:00:29.568 --> 00:00:31.878
P of x.

00:00:31.878 --> 00:00:35.941
All right, so I'm assuming
you have attempted this.

00:00:35.941 --> 00:00:38.357
We just have to realize,
okay, if x minus five

00:00:38.357 --> 00:00:40.562
is a factor of p of x,
that means you could

00:00:40.562 --> 00:00:44.915
write p of x, let me do
this in that green color,

00:00:44.915 --> 00:00:48.863
that means you could write p of x as being

00:00:48.863 --> 00:00:53.863
equal to x minus five
times some other business,

00:00:55.596 --> 00:00:57.662
times some other polynomial.

00:00:57.662 --> 00:01:01.342
So times some other polynomial.

00:01:01.342 --> 00:01:04.178
I don't know, let's just
call it g of x maybe,

00:01:04.178 --> 00:01:08.394
g of x, and this would be
a 2nd degree polynomial.

00:01:08.394 --> 00:01:11.052
So what would p of five have to be?

00:01:11.052 --> 00:01:12.805
Well p of five would have to zero.

00:01:12.805 --> 00:01:15.092
Five would have to be a
root of this polynomial.

00:01:15.092 --> 00:01:16.427
You see it right over here.

00:01:16.427 --> 00:01:19.988
If you replace this with the five,

00:01:20.158 --> 00:01:21.910
then this is going to be a five,

00:01:21.910 --> 00:01:22.955
and this is going to be a five.

00:01:22.955 --> 00:01:24.719
It doesn't matter what g of five is.

00:01:24.719 --> 00:01:26.425
This is going to be zero.

00:01:26.520 --> 00:01:31.520
So if x minus five is a factor,

00:01:37.422 --> 00:01:42.422
if and only if p of five is equal to zero.

00:01:46.723 --> 00:01:48.802
You could say that five is a root

00:01:48.802 --> 00:01:50.984
of the polynomial p.

00:01:50.984 --> 00:01:54.070
P of five is equal to zero.

00:01:54.084 --> 00:01:57.079
So let's just set p of five equal to zero

00:01:57.079 --> 00:01:58.644
and then try to solve for c.

00:01:58.947 --> 00:02:00.688
All right, so we're going to get five

00:02:00.688 --> 00:02:01.774
to the 3rd power.

00:02:01.774 --> 00:02:04.176
So this right over here, five
to the 3rd, so p of five.

00:02:04.944 --> 00:02:06.998
P, let me just write it down, p of five

00:02:06.998 --> 00:02:10.480
is equal to five to the 3rd is 125

00:02:10.480 --> 00:02:13.572
plus two times five squared.

00:02:13.572 --> 00:02:16.660
So that's two times 25, plus 50,

00:02:16.660 --> 00:02:18.320
plus c times five,

00:02:18.320 --> 00:02:21.060
plus five c plus 10.

00:02:21.060 --> 00:02:23.034
That needs to be equal to zero.

00:02:23.034 --> 00:02:28.034
Now let's see, if we were
to add 125 to 50 to 10,

00:02:30.596 --> 00:02:33.277
we are going to get 175, 185.

00:02:33.277 --> 00:02:38.277
So we get 185 plus five
c is equal to zero.

00:02:40.486 --> 00:02:42.448
Subtract 185 from both sides,

00:02:42.448 --> 00:02:46.261
you get five c is equal to -185,

00:02:48.369 --> 00:02:53.369
or c is equal to -185 over five,

00:02:54.081 --> 00:02:56.345
which is going to be, let's see,

00:02:56.345 --> 00:03:01.278
it's -185 over five, which is equal to,

00:03:01.278 --> 00:03:05.320
let's see, five goes into 180,

00:03:05.320 --> 00:03:07.345
let's see, it goes into, well it's five,

00:03:07.345 --> 00:03:09.121
let's see, it's going to be 30.

00:03:09.121 --> 00:03:12.523
Five times 30 is 150.

00:03:14.030 --> 00:03:16.422
Then we'll have another 35 to go,

00:03:16.422 --> 00:03:19.038
so it's going to be 37, -37.

00:03:19.038 --> 00:03:19.750
Is that right?

00:03:19.750 --> 00:03:23.447
Five times 30 is 150,
five times seven is 35,

00:03:23.447 --> 00:03:25.743
yep, -37.

00:03:26.113 --> 00:03:26.843
And we're done.

00:03:26.843 --> 00:03:29.362
If this was x to the 3rd plus 2 x squared

00:03:29.362 --> 00:03:33.507
minus 37 x plus 10, then x minus five

00:03:33.507 --> 00:03:36.047
would be a factor of this polynomial.


Polynomial remainder theorem example

WEBVTT
Kind: captions
Language: en

00:00:00.452 --> 00:00:02.228
- So we have a polynomial here.

00:00:02.228 --> 00:00:04.848
What I'm curious about
is what is the remainder

00:00:04.848 --> 00:00:08.133
if I were to divide this polynomial by,

00:00:08.133 --> 00:00:13.133
let's just say, x minus,
I want the remainder

00:00:15.690 --> 00:00:19.102
when I divide this
polynomial by x minus two?

00:00:19.933 --> 00:00:21.017
You could do this.

00:00:21.017 --> 00:00:22.921
You could figure this out with algebraic

00:00:22.921 --> 00:00:24.580
long division, but I'll give you a hint.

00:00:24.580 --> 00:00:26.798
It is much simpler and
much less computation

00:00:26.798 --> 00:00:28.679
intensive and takes much less space

00:00:28.679 --> 00:00:30.397
on your paper if you use

00:00:30.397 --> 00:00:32.256
the polynomial remainder theorem.

00:00:32.256 --> 00:00:34.775
If that's unfamiliar to
you, there's other videos

00:00:34.775 --> 00:00:36.425
that actually cover that.

00:00:36.425 --> 00:00:38.259
So why don't you have a go at it.

00:00:38.259 --> 00:00:41.409
All right, so now let's
work through this together.

00:00:41.409 --> 00:00:43.147
The polynomial remainder theorem tells us

00:00:43.147 --> 00:00:48.147
that when I take a polynomial, p of x,

00:00:50.484 --> 00:00:55.484
and if I were to divide
it by an x minus a,

00:00:56.578 --> 00:01:01.578
the remainder of that is just going

00:01:01.687 --> 00:01:03.904
to be equal to p of a.

00:01:04.427 --> 00:01:09.427
Is just going to be equal to p of a.

00:01:10.458 --> 00:01:12.535
So in this case, our p of x is this.

00:01:12.535 --> 00:01:13.825
What is our a?

00:01:13.825 --> 00:01:15.863
Well our a is going to be positive two.

00:01:15.863 --> 00:01:18.347
Remember it's x minus a.

00:01:18.347 --> 00:01:19.593
So let me do this.

00:01:19.593 --> 00:01:23.259
Our a is equal to positive two.

00:01:23.413 --> 00:01:24.307
So to figure out the remainder,

00:01:24.307 --> 00:01:27.354
we just have to evaluate p of two.

00:01:27.354 --> 00:01:28.389
So let's do that.

00:01:28.900 --> 00:01:32.488
So the remainder in this case

00:01:32.488 --> 00:01:37.488
is going to be equal to p of two,

00:01:40.905 --> 00:01:44.225
which is equal to, so let's see,

00:01:44.225 --> 00:01:46.419
it's going to be, I'll
just do it all in magenta,

00:01:46.419 --> 00:01:51.419
negative three times
eight minus, let's see,

00:01:52.038 --> 00:01:57.038
minus four times four plus 20 minus seven.

00:01:58.872 --> 00:02:03.810
So let's see, this is -24 minus 16

00:02:03.810 --> 00:02:08.349
plus 20 minus seven.

00:02:09.001 --> 00:02:13.575
So that gives us, let's see, -24 minus 16,

00:02:13.575 --> 00:02:15.315
this is -40.

00:02:15.810 --> 00:02:17.250
All right, I'm just doing it step by step.

00:02:17.250 --> 00:02:19.212
This is equal to negative,
actually I can do

00:02:19.212 --> 00:02:20.264
this in my head.

00:02:20.264 --> 00:02:21.437
All right, here we go.

00:02:21.437 --> 00:02:26.437
So this is -40 plus 20 is
-20 minus seven is -27.

00:02:30.435 --> 00:02:35.232
That was pretty neat
because if we attempted

00:02:35.232 --> 00:02:38.530
to do this without the
polynomial remainder theorem,

00:02:38.530 --> 00:02:40.630
we would have had to do a bunch

00:02:40.630 --> 00:02:41.753
of algebraic long division.

00:02:41.753 --> 00:02:43.342
Now if we did the algebraic long division,

00:02:43.342 --> 00:02:44.979
we would have gotten the
quotient and all of that,

00:02:44.979 --> 00:02:47.022
but we don't need the quotient,

00:02:47.022 --> 00:02:48.543
we don't need to know.

00:02:48.543 --> 00:02:50.482
So if we did all the
algebraic long division,

00:02:50.482 --> 00:02:53.337
you know, we would have taken our p of x

00:02:53.337 --> 00:02:55.938
and then we would have
divided the x minus a

00:02:55.938 --> 00:03:00.938
into it, and we would have
gotten a quotient here,

00:03:03.495 --> 00:03:05.202
q of x, and we would have done all this

00:03:05.202 --> 00:03:07.399
business down here, all this
algebraic long division.

00:03:07.399 --> 00:03:09.234
Probably wouldn't have
even fit on the page.

00:03:09.234 --> 00:03:11.671
But eventually we would
have gotten to a point

00:03:11.671 --> 00:03:12.798
where we got an expression that has

00:03:12.798 --> 00:03:14.098
a lower degree than this.

00:03:14.098 --> 00:03:15.650
It would have to be a
constant because this is

00:03:15.650 --> 00:03:17.262
a 1st degree, so it
would have be essentially

00:03:17.262 --> 00:03:18.550
a zero degree.

00:03:18.550 --> 00:03:21.849
So we would have eventually
gotten to our -27.

00:03:21.849 --> 00:03:24.217
But this was much, much, much, much easier

00:03:24.217 --> 00:03:28.199
then having to go through
this entire exercise.

00:03:28.199 --> 00:03:30.020
Hopefully you appreciated that.