Example: Combinatorics and probability

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.950
A card game using 36 unique
cards, four suits, diamonds,

00:00:04.950 --> 00:00:08.500
hearts, clubs, and spades, with
cards numbered from 1 to

00:00:08.500 --> 00:00:09.410
9 in each suit.

00:00:09.410 --> 00:00:10.440
So there's four suits.

00:00:10.440 --> 00:00:12.220
Each of them have nine
cards, so that gives

00:00:12.220 --> 00:00:14.330
us 36 unique cards.

00:00:14.330 --> 00:00:17.980
A hand is a collection of nine
cards, which can be sorted

00:00:17.980 --> 00:00:19.310
however the player chooses.

00:00:19.310 --> 00:00:22.000
So they're essentially telling
us that order does not matter.

00:00:22.000 --> 00:00:26.900
What is the probability of
getting all four of the 1's?

00:00:26.900 --> 00:00:31.900
So they want to know the
probability of getting all

00:00:31.900 --> 00:00:33.670
four of the 1's.

00:00:33.670 --> 00:00:43.340
So all four 1's in
my hand of 9.

00:00:43.340 --> 00:00:45.850
Now this is kind of daunting at
first. You're like, gee you

00:00:45.850 --> 00:00:48.160
know, I've got nine cards and
I'm taking them out of 36 and

00:00:48.160 --> 00:00:50.200
I have to figure out how do
I get all of the 1's.

00:00:50.200 --> 00:00:53.240
But if we think about it just
very, very, in very simple

00:00:53.240 --> 00:00:58.400
terms, all a probability is
saying is, the number of

00:00:58.400 --> 00:01:01.660
events-- or I guess you could
say-- the number of ways in

00:01:01.660 --> 00:01:05.030
which this action or
this event happens.

00:01:05.030 --> 00:01:06.960
So this is what the definition
of the probability is.

00:01:06.960 --> 00:01:18.210
It's going to be the number of
ways in which event can happen

00:01:18.210 --> 00:01:20.910
and when we talk about the
event, we're talking about

00:01:20.910 --> 00:01:23.270
having all four 1's
in my hand.

00:01:23.270 --> 00:01:24.480
That's the event.

00:01:24.480 --> 00:01:27.810
And all of these different ways,
that's sometimes called

00:01:27.810 --> 00:01:28.740
the event space.

00:01:28.740 --> 00:01:32.100
But we actually want to count
how many ways that, if I get a

00:01:32.100 --> 00:01:35.890
hand of 9 picking from
36, that I can get

00:01:35.890 --> 00:01:37.120
the four 1's in it.

00:01:37.120 --> 00:01:39.730
So it is the number of ways in
which my event can happen and

00:01:39.730 --> 00:01:48.480
we want to divide that into all
of the possibilities-- or

00:01:48.480 --> 00:01:51.220
maybe I should write it this
way-- the total number of

00:01:51.220 --> 00:01:52.470
hands that I can get.

00:01:58.520 --> 00:02:03.900
So the numerator in blue is the
number of different hands

00:02:03.900 --> 00:02:08.639
where I have the four 1's and
we're dividing the total

00:02:08.639 --> 00:02:09.449
number of hands.

00:02:09.449 --> 00:02:11.620
Now let's figure out the total
number of hands first, because

00:02:11.620 --> 00:02:13.360
on some level this might be
more intuitive and we've

00:02:13.360 --> 00:02:15.290
actually done this before.

00:02:15.290 --> 00:02:17.040
Now, the total number
of hands, we're

00:02:17.040 --> 00:02:18.840
picking nine cards.

00:02:18.840 --> 00:02:21.900
And we're picking them from
a set of 36 unique cards.

00:02:21.900 --> 00:02:23.510
And we've done this
many, many times.

00:02:23.510 --> 00:02:29.500
Let me write this, total number
of hands, or total

00:02:29.500 --> 00:02:31.420
number of possible hands.

00:02:31.420 --> 00:02:34.780
That's equal to-- you can
imagine, you have nine cards

00:02:34.780 --> 00:02:35.750
to pick from.

00:02:35.750 --> 00:02:39.010
The first card you pick, it's
going to be 1 of 36 cards.

00:02:39.010 --> 00:02:41.220
Then the next one is going
to be 1 of 35.

00:02:41.220 --> 00:02:46.130
Then the next one is going to
be 1 of 34, 33, 32, 31.

00:02:46.130 --> 00:02:48.550
We're going to do this nine
times, one, two, three, four,

00:02:48.550 --> 00:02:55.160
five, six, seven,
eight, and nine.

00:02:55.160 --> 00:02:57.375
So that would be the
total number of

00:02:57.375 --> 00:02:59.000
hands if order mattered.

00:02:59.000 --> 00:03:00.800
But we know-- and we've gone
over this before-- that we

00:03:00.800 --> 00:03:02.500
don't care about the order.

00:03:02.500 --> 00:03:04.650
All we care about are the actual
cars that are in there.

00:03:04.650 --> 00:03:06.310
So we're overcounting here.

00:03:06.310 --> 00:03:08.850
We're overcounting for all of
the different rearrangements

00:03:08.850 --> 00:03:13.550
that these cards could have. It
doesn't matter whether the

00:03:13.550 --> 00:03:15.540
Ace of diamonds is the
first card I pick or

00:03:15.540 --> 00:03:16.730
the last card I pick.

00:03:16.730 --> 00:03:18.980
The way I've counted them right
now, we are counting

00:03:18.980 --> 00:03:21.110
those as two separate hands.

00:03:21.110 --> 00:03:22.770
But they aren't two
separate hands, so

00:03:22.770 --> 00:03:23.560
order doesn't matter.

00:03:23.560 --> 00:03:28.500
So we have to do is, we have to
divide this by the number

00:03:28.500 --> 00:03:30.600
of ways you can arrange
nine things.

00:03:30.600 --> 00:03:33.800
So you could put nine of the
things in the first position,

00:03:33.800 --> 00:03:36.500
then eight in the second,
seven in the third, so

00:03:36.500 --> 00:03:37.950
forth and so on.

00:03:37.950 --> 00:03:41.450
It essentially becomes 9
factorial times 2 times 1.

00:03:41.450 --> 00:03:42.920
And we've seen this
multiple times.

00:03:42.920 --> 00:03:45.400
This is essentially
36 choose 9.

00:03:45.400 --> 00:03:47.700
This expression right here is
the same thing-- just you can

00:03:47.700 --> 00:03:51.880
relate it to the combinatorics
formulas that you might be

00:03:51.880 --> 00:03:55.470
familiar with-- this is the
same thing as 36 factorial

00:03:55.470 --> 00:04:00.480
over 36 minus 9 factorial--
that's what this orange part

00:04:00.480 --> 00:04:05.930
is over here-- divided by 9
factorial or over 9 factorial.

00:04:05.930 --> 00:04:09.410
What's green is what's green and
what is orange is what's

00:04:09.410 --> 00:04:09.980
orange there.

00:04:09.980 --> 00:04:11.990
So that's the total
number of hands.

00:04:11.990 --> 00:04:14.730
Now a little bit more of a
nuanced thought process is,

00:04:14.730 --> 00:04:16.720
how do we figure out the number
of ways in which the

00:04:16.720 --> 00:04:19.695
event can happen, in which
we can have all four 1's.

00:04:19.695 --> 00:04:21.680
So let's figure that out.

00:04:21.680 --> 00:04:28.050
So number of ways-- or maybe we
should say this-- number of

00:04:28.050 --> 00:04:34.355
hands with four 1's.

00:04:37.420 --> 00:04:39.310
And just as a little bit of a
thought experiment, imagine if

00:04:39.310 --> 00:04:41.600
we were only taking four cards,
if a hand only had four

00:04:41.600 --> 00:04:42.710
cards in it.

00:04:42.710 --> 00:04:48.270
Well if a hand only had four
cards in it, then the number

00:04:48.270 --> 00:04:52.240
of ways to get a hand with four
1's, there'd only be one

00:04:52.240 --> 00:04:53.950
way, one combination.

00:04:53.950 --> 00:04:55.200
You'd just have four 1's.

00:04:55.200 --> 00:04:57.550
That's the only combination with
four 1's, if we were only

00:04:57.550 --> 00:04:59.130
picking four cards.

00:04:59.130 --> 00:05:03.010
But here, we're not only
picking four cards.

00:05:03.010 --> 00:05:08.210
Four of the cards are
going to be 1's.

00:05:08.210 --> 00:05:10.480
One, two, three, four.

00:05:10.480 --> 00:05:13.660
But the other five cards are
going to be different.

00:05:13.660 --> 00:05:17.210
So one, two, three,
four, five.

00:05:17.210 --> 00:05:22.460
So for the other five cards--
if you imagine this slot--

00:05:22.460 --> 00:05:26.880
considering that of the 36 we
would have to pick four of

00:05:26.880 --> 00:05:32.640
them already in order for
us to have four 1's.

00:05:32.640 --> 00:05:35.260
Well, we've used up four of
them, so there's 32 possible

00:05:35.260 --> 00:05:38.500
cards over in that position
of the hand.

00:05:38.500 --> 00:05:41.750
And then there'd be 31 in that
position of the hand.

00:05:41.750 --> 00:05:44.480
And then there'd be 30 because
every time we're picking a

00:05:44.480 --> 00:05:45.400
card, were using it up.

00:05:45.400 --> 00:05:46.720
And now we only have
30 to pick from.

00:05:46.720 --> 00:05:50.440
Then we only have
29 to pick from.

00:05:50.440 --> 00:05:52.820
And then we have 28
to pick from.

00:05:52.820 --> 00:05:56.020
And just like we did before,
we don't care about order.

00:05:56.020 --> 00:05:59.055
We don't care if we pick the 5
of clubs first or whether we

00:05:59.055 --> 00:06:01.680
pick the 5 of clubs last. So we
shouldn't double count it.

00:06:01.680 --> 00:06:04.380
So we have to divide by the
different number of ways that

00:06:04.380 --> 00:06:06.670
five cards can be arranged.

00:06:06.670 --> 00:06:12.310
So we have to divide this by the
different ways that five

00:06:12.310 --> 00:06:13.190
cards can be arranged.

00:06:13.190 --> 00:06:15.890
The first card or the first
position can be any one of

00:06:15.890 --> 00:06:19.210
five cards, then four cards,
then three cards, then two

00:06:19.210 --> 00:06:20.830
cards, then one cards.

00:06:20.830 --> 00:06:23.040
So the number of hands
with four 1's is

00:06:23.040 --> 00:06:26.060
actually just this number.

00:06:26.060 --> 00:06:28.400
You're actually looking at all
of the different ways you can

00:06:28.400 --> 00:06:29.840
fill up the remaining cards.

00:06:29.840 --> 00:06:31.350
These four 1's are just
going to be four 1's.

00:06:31.350 --> 00:06:34.060
There's only one way to get that
if the remaining cards

00:06:34.060 --> 00:06:36.810
that's going to give all of the
different combinations of

00:06:36.810 --> 00:06:37.900
having four 1's.

00:06:37.900 --> 00:06:40.350
So this will be a count of all
of the different combinations

00:06:40.350 --> 00:06:42.780
because all of the different
extra stuff that you have will

00:06:42.780 --> 00:06:44.520
be all of the different hands.

00:06:44.520 --> 00:06:46.930
Now we know the total
number of hands with

00:06:46.930 --> 00:06:48.550
four 1's is this number.

00:06:48.550 --> 00:06:50.010
And now we can divide
it by the total

00:06:50.010 --> 00:06:51.210
number of possible hands.

00:06:51.210 --> 00:06:53.280
And I didn't multiply them out
on purpose so that we can

00:06:53.280 --> 00:06:54.240
cancel things out.

00:06:54.240 --> 00:06:55.530
So let's do that.

00:06:55.530 --> 00:06:57.950
Let's take this and
divide by that.

00:06:57.950 --> 00:06:59.460
So let me just copy
and paste it.

00:07:06.140 --> 00:07:10.870
Let's take that and let's
divide it by that.

00:07:10.870 --> 00:07:13.780
But dividing by a fraction is
the same thing as multiplying

00:07:13.780 --> 00:07:14.820
by the reciprocal.

00:07:14.820 --> 00:07:17.250
So let's just multiply
by the reciprocal.

00:07:17.250 --> 00:07:19.350
So let's multiply-- so this
is the denominator.

00:07:19.350 --> 00:07:22.300
Let's make this the numerator.

00:07:22.300 --> 00:07:24.900
So let me copy it and then
let me paste it.

00:07:24.900 --> 00:07:27.440
So that's the numerator
and then that's the

00:07:27.440 --> 00:07:28.900
denominator up there.

00:07:28.900 --> 00:07:31.720
Because we're dividing
by that expression.

00:07:31.720 --> 00:07:37.110
So let me-- whoops.

00:07:37.110 --> 00:07:39.050
Let me put that there.

00:07:39.050 --> 00:07:42.590
Let me get the select tool and
then let me make sure I'm

00:07:42.590 --> 00:07:45.110
selecting all of the numbers.

00:07:45.110 --> 00:07:47.970
Let me copy it and then
let me paste that.

00:07:47.970 --> 00:07:49.880
It's a little messy with those
lines there, but I think

00:07:49.880 --> 00:07:52.070
this'll suit our purposes.

00:07:52.070 --> 00:07:56.260
This'll suit our purposes
just fine.

00:07:56.260 --> 00:07:57.890
So when we're multiplying by
this, we're essentially

00:07:57.890 --> 00:08:00.130
dividing by this expression
up here.

00:08:00.130 --> 00:08:03.250
Now this we can simplify
pretty easily.

00:08:03.250 --> 00:08:07.820
We have a-- well actually I
forgot to do-- this should be

00:08:07.820 --> 00:08:08.610
9 factorial.

00:08:08.610 --> 00:08:12.070
This should be 9 times 8 times
7 times 6 times 5 times 4

00:08:12.070 --> 00:08:13.570
times 3 times 2 times 1.

00:08:13.570 --> 00:08:15.260
Let me put that in
both places.

00:08:15.260 --> 00:08:20.190
Actually let me just-- let me
clear that both places.

00:08:20.190 --> 00:08:20.620
Clear.

00:08:20.620 --> 00:08:23.470
Don't want to confuse people.

00:08:23.470 --> 00:08:25.520
Clear.

00:08:25.520 --> 00:08:27.280
I'm sorry if that confused you
when I wrote it earlier.

00:08:27.280 --> 00:08:28.260
This would be 9 factorial.

00:08:28.260 --> 00:08:33.190
9 times 8 times 7 times 6
times 5 times 4 times 3

00:08:33.190 --> 00:08:35.230
times 2 times 1.

00:08:35.230 --> 00:08:38.760
Let me copy and paste
that now.

00:08:38.760 --> 00:08:41.230
Copy and then you paste.

00:08:41.230 --> 00:08:44.190
It That's that, right there.

00:08:44.190 --> 00:08:45.960
And then we have this
in the numerator.

00:08:45.960 --> 00:08:48.720
We have 5 times 4 times 3 times
1 in the denominator.

00:08:48.720 --> 00:08:53.400
So this will cancel out with
that part right over there.

00:08:53.400 --> 00:08:58.980
And then we have 32 times 31
times 30 times 29 times 28.

00:08:58.980 --> 00:09:01.150
That is going to cancel
with that.

00:09:01.150 --> 00:09:04.230
That and that cancels out.

00:09:04.230 --> 00:09:06.630
So what we're left with is
just this part over here.

00:09:06.630 --> 00:09:07.650
Let me rewrite it.

00:09:07.650 --> 00:09:15.710
So we're left with 9 times 8
times 7 times 6 over-- and

00:09:15.710 --> 00:09:18.380
this will just be an exercise
in simplifying this

00:09:18.380 --> 00:09:23.920
expression-- 36 times 35
times 34 times 33.

00:09:23.920 --> 00:09:26.360
And let's see, if we divide the
numerator and denominator

00:09:26.360 --> 00:09:29.550
by 9, that becomes a 1,
this becomes a 4.

00:09:29.550 --> 00:09:31.780
You can divide the numerator
and denominator by 4, this

00:09:31.780 --> 00:09:32.710
becomes a 2.

00:09:32.710 --> 00:09:35.680
This becomes a 1.

00:09:35.680 --> 00:09:39.470
You divide numerator and
denominator by 7, this becomes

00:09:39.470 --> 00:09:42.530
a 1, this becomes a 5.

00:09:42.530 --> 00:09:46.170
You can divide both by 2 again
and then this becomes a 1.

00:09:46.170 --> 00:09:48.140
This becomes a 17.

00:09:48.140 --> 00:09:50.500
And you could divide
this and this by 3.

00:09:50.500 --> 00:09:55.260
This becomes a 2 and then
this becomes an 11.

00:09:55.260 --> 00:09:59.550
So we're left with, the
probability of having all four

00:09:59.550 --> 00:10:04.100
1's in my hand of 9 that I'm
selecting from 36 unique cards

00:10:04.100 --> 00:10:05.970
is equal to-- in the numerator,
I'm just left with

00:10:05.970 --> 00:10:10.880
this 2 times 1 times 1 times 1--
so it's equal to 2 over 5

00:10:10.880 --> 00:10:15.150
times 17 times 11.

00:10:15.150 --> 00:10:19.980
And that is-- so drum roll, this
was kind of an involved

00:10:19.980 --> 00:10:28.660
problem-- 5 times 17 times
11 is equal to 935.

00:10:28.660 --> 00:10:34.150
So it's equal to 2 over 935.

00:10:34.150 --> 00:10:37.800
So about roughly 2 in a thousand
chance or 1 in a

00:10:37.800 --> 00:10:41.040
500-- roughly speaking, this
isn't exact odds-- you have a

00:10:41.040 --> 00:10:45.020
roughly 1 in 500 chance of
getting all four of the 1's in

00:10:45.020 --> 00:10:47.000
your hand of 9 when
you're selecting

00:10:47.000 --> 00:10:50.190
from 36 unique cards.


Example: Lottery probability

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.390
To win a particular lottery
game, a player chooses 4

00:00:04.390 --> 00:00:06.490
numbers from 1 to 60.

00:00:06.490 --> 00:00:09.490
Each number can only
be chosen once.

00:00:09.490 --> 00:00:14.040
If all 4 numbers match the 4
winning numbers, regardless of

00:00:14.040 --> 00:00:16.149
order, the player wins.

00:00:16.149 --> 00:00:20.340
What is the probability that the
winning numbers are 3, 15,

00:00:20.340 --> 00:00:22.550
46, and 49?

00:00:22.550 --> 00:00:24.920
So the way to think about this
problem, they say that we're

00:00:24.920 --> 00:00:28.330
going to choose four
numbers from 60.

00:00:28.330 --> 00:00:30.600
So one way to think about it
is, how many different

00:00:30.600 --> 00:00:35.230
outcomes are there if we choose
four numbers out of 60?

00:00:35.230 --> 00:00:38.610
Now this is equivalent to
saying, how many combinations

00:00:38.610 --> 00:00:41.080
are there if we have 60 items?

00:00:41.080 --> 00:00:43.990
In this case we have 60 numbers,
and we are going to

00:00:43.990 --> 00:00:46.770
choose four.

00:00:46.770 --> 00:00:49.050
And we don't care
about the order.

00:00:49.050 --> 00:00:51.170
That's why we're dealing
with combinations, not

00:00:51.170 --> 00:00:51.900
permutations.

00:00:51.900 --> 00:00:53.660
We don't care about the order.

00:00:53.660 --> 00:00:57.310
So how many different groups of
four can we pick out of 60?

00:00:57.310 --> 00:00:59.460
And we don't care what order
we picked them in.

00:00:59.460 --> 00:01:02.810
And we've seen in previous
videos that there is a formula

00:01:02.810 --> 00:01:04.920
here, but it's important to
understand the reasoning

00:01:04.920 --> 00:01:05.459
behind the formula.

00:01:05.459 --> 00:01:08.210
I'll write the formula here,
but we'll think about what

00:01:08.210 --> 00:01:09.660
it's actually saying.

00:01:09.660 --> 00:01:18.560
So this is 60 factorial over 60
minus 4 factorial, divided

00:01:18.560 --> 00:01:21.080
also by 4 factorial,
or the denominator

00:01:21.080 --> 00:01:22.620
multiplied by 4 factorial.

00:01:22.620 --> 00:01:24.135
So this is the formula
right here.

00:01:24.135 --> 00:01:28.370
But what this is really saying,
this part right here,

00:01:28.370 --> 00:01:33.640
60 factorial divided by 60 minus
4 factorial, that's just

00:01:33.640 --> 00:01:41.720
60 times 59, times
58, times 57.

00:01:41.720 --> 00:01:44.100
That's what this expression
right here is.

00:01:44.100 --> 00:01:47.050
And if you think about it, the
first number you pick--

00:01:47.050 --> 00:01:50.170
there's 1 of 60 numbers, but
then that number is kind of

00:01:50.170 --> 00:01:51.100
out of the game.

00:01:51.100 --> 00:01:54.340
Then you can pick from 1 of
59, then from 1 of 58,

00:01:54.340 --> 00:01:55.840
then of 1 of 57.

00:01:55.840 --> 00:02:00.010
So if you cared about order,
this is the number of

00:02:00.010 --> 00:02:00.830
permutations.

00:02:00.830 --> 00:02:04.620
You could pick four items out of
60 without replacing them.

00:02:04.620 --> 00:02:07.860
Now, this is when you cared
about order, but you're

00:02:07.860 --> 00:02:10.169
overcounting because it's
counting different

00:02:10.169 --> 00:02:11.930
permutations that are
essentially the same

00:02:11.930 --> 00:02:15.760
combination, essentially the
same set of four numbers.

00:02:15.760 --> 00:02:19.220
And that's why we're dividing
by 4 factorial here.

00:02:19.220 --> 00:02:22.840
Because 4 factorial is
essentially the number of ways

00:02:22.840 --> 00:02:26.250
that four numbers can be
arranged in four places.

00:02:26.250 --> 00:02:26.530
Right?

00:02:26.530 --> 00:02:28.800
The first number can be in one
of four slots, the second in

00:02:28.800 --> 00:02:30.860
one of three, then
two, then one.

00:02:30.860 --> 00:02:33.090
That's why you're dividing
by 4 factorial.

00:02:33.090 --> 00:02:34.830
But anyway, let's just
evaluate this.

00:02:34.830 --> 00:02:38.200
This'll tell us how many
possible outcomes are there

00:02:38.200 --> 00:02:39.240
for the lottery game.

00:02:39.240 --> 00:02:44.150
So this is equal to-- we already
said the blue part is

00:02:44.150 --> 00:02:52.110
equivalent to 60 times 59,
times 58, times 57.

00:02:52.110 --> 00:02:54.540
So that's literally 60
factorial divided by

00:02:54.540 --> 00:02:56.410
essentially 56 factorial.

00:02:56.410 --> 00:02:59.210
And then you have your 4
factorial over here, which is

00:02:59.210 --> 00:03:03.320
4 times 3, times 2, times 1.

00:03:03.320 --> 00:03:05.830
And we could simplify it a
little bit just before we

00:03:05.830 --> 00:03:07.150
break out the calculator.

00:03:07.150 --> 00:03:11.020
60 divided by 4 is 15.

00:03:11.020 --> 00:03:14.235
And then let's see, 15
divided by 3 is 5.

00:03:17.650 --> 00:03:23.910
And let's see, we have a
58 divided by 2 is 29.

00:03:23.910 --> 00:03:34.460
So our answer is going to be 5
times 59, times 29, times 57.

00:03:34.460 --> 00:03:35.520
Now this isn't going
to be our answer.

00:03:35.520 --> 00:03:38.720
This is going to be the number
of combinations we can get if

00:03:38.720 --> 00:03:41.950
we choose four numbers
out of 60 and we

00:03:41.950 --> 00:03:43.090
don't care about order.

00:03:43.090 --> 00:03:45.710
So let's take the calculator
out now.

00:03:45.710 --> 00:03:54.950
So we have 5 times 59,
times 29, times 57.

00:03:54.950 --> 00:04:00.590
It's equal to 487,635.

00:04:00.590 --> 00:04:04.720
So let me write that down.

00:04:04.720 --> 00:04:16.450
That is 487,635 combinations.

00:04:16.450 --> 00:04:18.810
If you're picking four numbers,
you're choosing four

00:04:18.810 --> 00:04:22.270
numbers out of 60, or
60 choose four.

00:04:22.270 --> 00:04:24.670
Now, the question they say is,
what is the probability that

00:04:24.670 --> 00:04:29.090
the winning numbers are
3, 15, 46, and 49?

00:04:29.090 --> 00:04:31.420
Well, this is just one
particular of the

00:04:31.420 --> 00:04:32.620
combinations.

00:04:32.620 --> 00:04:40.850
This is just one of the 487,635
possible outcomes.

00:04:40.850 --> 00:04:51.020
So the probability of 3, 15, 46,
49 winning is just equal

00:04:51.020 --> 00:04:58.670
to-- well, this is just one of
the outcomes out of 487,635.

00:04:58.670 --> 00:05:01.890
So that right there is your
probability of winning.

00:05:01.890 --> 00:05:04.860
This is one outcome out of all
the potential outcomes or

00:05:04.860 --> 00:05:09.070
combinations when you take 60
and you choose four from that.


Example: Different ways to pick officers

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.010
A club of nine people wants
to choose a board of three

00:00:04.010 --> 00:00:07.600
officers, President, Vice
President, and Secretary.

00:00:07.600 --> 00:00:10.520
Assuming the officers are chosen
at random, what is the

00:00:10.520 --> 00:00:13.390
probability that the officers
are Marcia for president,

00:00:13.390 --> 00:00:17.300
Sabita for Vice President,
and Robert for Secretary?

00:00:17.300 --> 00:00:22.510
So to think about the
probability of Marcia-- so let

00:00:22.510 --> 00:00:29.795
me write this-- President is
equal to Marcia, or Vice

00:00:29.795 --> 00:00:34.500
President is equal
to Sabita, and

00:00:34.500 --> 00:00:37.375
Secretary is equal to Robert.

00:00:40.980 --> 00:00:43.630
This, right here, is one
possible outcome,

00:00:43.630 --> 00:00:44.705
one specific outcome.

00:00:44.705 --> 00:00:49.240
So it's one outcome out of the
total number of outcomes, over

00:00:49.240 --> 00:00:52.915
the total number of
possibilities.

00:00:55.940 --> 00:00:59.000
Now what is the total number
of possibilities?

00:00:59.000 --> 00:01:00.670
Well to think about that, let's
just think about the

00:01:00.670 --> 00:01:03.980
three positions.

00:01:03.980 --> 00:01:06.920
You have President, you
have Vice President,

00:01:06.920 --> 00:01:08.670
and you have Secretary.

00:01:08.670 --> 00:01:10.860
Now let's just assume that we're
going to fill the slot

00:01:10.860 --> 00:01:13.060
of President first. We don't
have to do President first,

00:01:13.060 --> 00:01:14.370
but we're just going
to pick here.

00:01:14.370 --> 00:01:16.940
So if we're just picking
President first, we haven't

00:01:16.940 --> 00:01:19.880
assigned anyone to any officers
just yet, so we have

00:01:19.880 --> 00:01:21.630
nine people to choose from.

00:01:21.630 --> 00:01:23.950
So there are nine possibilities
here.

00:01:23.950 --> 00:01:26.530
Now, when we go to selecting our
Vice President, we would

00:01:26.530 --> 00:01:29.330
have already assigned one
person to the President.

00:01:29.330 --> 00:01:31.740
So we only have eight
people to pick from.

00:01:31.740 --> 00:01:34.100
And when we assign our
Secretary, we would've already

00:01:34.100 --> 00:01:36.570
assigned our President and Vice
President, so we're only

00:01:36.570 --> 00:01:38.830
going to have seven people
to pick from.

00:01:38.830 --> 00:01:42.190
So the total permutations here
or the total number of

00:01:42.190 --> 00:01:44.850
possibilities, or the total
number of ways, to pick

00:01:44.850 --> 00:01:47.820
President, Vice President, and
Secretary from nine people, is

00:01:47.820 --> 00:01:50.990
going to be 9 times 8 times 7.

00:01:50.990 --> 00:01:54.310
Which is, let's see,
9 times 8 is 72.

00:01:54.310 --> 00:02:01.890
72 times 7, 2 times 7 is 14, 7
times 7 is 49 plus 1 is 50.

00:02:01.890 --> 00:02:07.620
So there's 504 possibilities.

00:02:07.620 --> 00:02:10.789
So to answer the question, the
probability of Marcia being

00:02:10.789 --> 00:02:13.080
President, Sabita being Vice
President, and Robert being

00:02:13.080 --> 00:02:15.880
Secretary is 1 over the total
number of possibilities, which

00:02:15.880 --> 00:02:19.960
is 1 over 504.

00:02:19.960 --> 00:02:21.780
That's the probability.


Example: 9 card hands

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:05.280
A card game using 36 unique
cards, four suits, diamonds,

00:00:05.280 --> 00:00:08.880
hearts, clubs and spades-- this
should be spades, not

00:00:08.880 --> 00:00:14.730
spaces-- with cards numbered
from 1 to 9 in each suit.

00:00:14.730 --> 00:00:16.190
A hand is chosen.

00:00:16.190 --> 00:00:20.910
A hand is a collection of 9
cards, which can be sorted

00:00:20.910 --> 00:00:22.410
however the player chooses.

00:00:22.410 --> 00:00:23.260
Fair enough.

00:00:23.260 --> 00:00:29.590
How many 9 card hands
are possible?

00:00:29.590 --> 00:00:30.300
So let's think about it.

00:00:30.300 --> 00:00:33.080
There are 36 unique cards-- and
I won't worry about, you

00:00:33.080 --> 00:00:36.990
know, there's nine numbers in
each suit, and there are four

00:00:36.990 --> 00:00:39.020
suits, 4 times 9 is 36.

00:00:39.020 --> 00:00:41.420
But let's just think of the
cards as being 1 through 36,

00:00:41.420 --> 00:00:43.330
and we're going to pick
nine of them.

00:00:43.330 --> 00:00:47.960
So at first we'll say, well
look, I have nine slots in my

00:00:47.960 --> 00:00:49.000
hand, right?

00:00:49.000 --> 00:00:54.460
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:00:54.460 --> 00:00:54.720
Right?

00:00:54.720 --> 00:00:57.430
I'm going to pick nine
cards for my hand.

00:00:57.430 --> 00:01:00.930
And so for the very first card,
how many possible cards

00:01:00.930 --> 00:01:02.130
can I pick from?

00:01:02.130 --> 00:01:04.769
Well, there's 36 unique cards,
so for that first slot,

00:01:04.769 --> 00:01:05.760
there's 36.

00:01:05.760 --> 00:01:07.680
But then that's now
part of my hand.

00:01:07.680 --> 00:01:09.710
Now for the second slot,
how many will there be

00:01:09.710 --> 00:01:10.980
left to pick from?

00:01:10.980 --> 00:01:12.390
Well, I've already picked
one, so there will only

00:01:12.390 --> 00:01:13.980
be 35 to pick from.

00:01:13.980 --> 00:01:16.010
And then for the third
slot, 34, and then

00:01:16.010 --> 00:01:16.800
it just keeps going.

00:01:16.800 --> 00:01:23.700
Then 33 to pick from, 32,
31, 30, 29, and 28.

00:01:23.700 --> 00:01:27.800
So you might want to say that
there are 36 times 35, times

00:01:27.800 --> 00:01:33.950
34, times 33, times 32, times
31, times 30, times 29, times

00:01:33.950 --> 00:01:36.020
28 possible hands.

00:01:36.020 --> 00:01:39.660
Now, this would be true
if order mattered.

00:01:39.660 --> 00:01:46.010
This would be true if
I have card 15 here.

00:01:46.010 --> 00:01:49.360
Maybe I have a-- let me put it
here-- maybe I have a 9 of

00:01:49.360 --> 00:01:53.040
spades here, and then I
have a bunch of cards.

00:01:53.040 --> 00:01:55.870
And maybe I have-- and
that's one hand.

00:01:55.870 --> 00:01:56.800
And then I have another.

00:01:56.800 --> 00:02:00.070
So then I have cards one,
two, three, four,

00:02:00.070 --> 00:02:01.230
five, six, seven, eight.

00:02:01.230 --> 00:02:02.350
I have eight other cards.

00:02:02.350 --> 00:02:05.500
Or maybe another hand is I have
the eight cards, 1, 2, 3,

00:02:05.500 --> 00:02:10.370
4, 5, 6, 7, 8, and then I
have the 9 of spades.

00:02:10.370 --> 00:02:13.440
If we were thinking of these
as two different hands,

00:02:13.440 --> 00:02:15.670
because we have the exact same
cards, but they're in

00:02:15.670 --> 00:02:18.390
different order, then what I
just calculated would make a

00:02:18.390 --> 00:02:20.850
lot of sense, because we
did it based on order.

00:02:20.850 --> 00:02:25.070
But they're telling us that
the cards can be sorted

00:02:25.070 --> 00:02:29.220
however the player chooses,
so order doesn't matter.

00:02:29.220 --> 00:02:30.900
So we're overcounting.

00:02:30.900 --> 00:02:34.270
We're counting all of the
different ways that the same

00:02:34.270 --> 00:02:36.810
number of cards can
be arranged.

00:02:36.810 --> 00:02:41.500
So in order to not overcount, we
have to divide this by the

00:02:41.500 --> 00:02:45.150
ways in which nine cards
can be rearranged.

00:02:45.150 --> 00:02:50.350
So we have to divide this by
the way nine cards can be

00:02:50.350 --> 00:02:51.160
rearranged.

00:02:51.160 --> 00:02:53.330
So how many ways can nine
cards be rearranged?

00:02:53.330 --> 00:02:56.520
If I have nine cards and I'm
going to pick one of nine to

00:02:56.520 --> 00:02:59.150
be in the first slot, well, that
means I have 9 ways to

00:02:59.150 --> 00:03:00.690
put something in
the first slot.

00:03:00.690 --> 00:03:04.050
Then in the second slot, I have
8 ways of putting a card

00:03:04.050 --> 00:03:06.710
in the second slot, because I
took one to put it in the

00:03:06.710 --> 00:03:08.170
first, so I have 8 left.

00:03:08.170 --> 00:03:16.390
Then 7, then 6, then 5, then
4, then 3, then 2, then 1.

00:03:16.390 --> 00:03:18.050
That last slot, there's only
going to be 1 card

00:03:18.050 --> 00:03:19.110
left to put in it.

00:03:19.110 --> 00:03:21.610
So this number right here,
where you take 9 times 8,

00:03:21.610 --> 00:03:24.530
times 7, times 6, times 5, times
4, times 3, times 2,

00:03:24.530 --> 00:03:27.610
times 1, or 9-- you start with
9 and then you multiply it by

00:03:27.610 --> 00:03:29.230
every number less than 9.

00:03:29.230 --> 00:03:32.340
Every, I guess we could say,
natural number less than 9.

00:03:32.340 --> 00:03:36.480
This is called 9 factorial,
and you express it as an

00:03:36.480 --> 00:03:37.660
exclamation mark.

00:03:37.660 --> 00:03:40.890
So if we want to think about all
of the different ways that

00:03:40.890 --> 00:03:43.640
we can have all of the different
combinations for

00:03:43.640 --> 00:03:47.160
hands, this is the number of
hands if we cared about the

00:03:47.160 --> 00:03:50.010
order, but then we want to
divide by the number of ways

00:03:50.010 --> 00:03:52.640
we can order things so that
we don't overcount.

00:03:52.640 --> 00:03:54.260
And this will be an answer
and this will

00:03:54.260 --> 00:03:55.230
be the correct answer.

00:03:55.230 --> 00:03:58.620
Now this is a super, super
duper large number.

00:03:58.620 --> 00:04:01.010
Let's figure out how large
of a number this is.

00:04:01.010 --> 00:04:11.060
We have 36-- let me scroll to
the left a little bit-- 36

00:04:11.060 --> 00:04:24.040
times 35, times 34, times 33,
times 32, times 31, times 30,

00:04:24.040 --> 00:04:29.870
times 29, times 28,
divided by 9.

00:04:29.870 --> 00:04:30.900
Well, I can do it this way.

00:04:30.900 --> 00:04:34.610
I can put a parentheses--
divided by parentheses, 9

00:04:34.610 --> 00:04:37.900
times 8, times 7, times 6, times
5, times 4, times 3,

00:04:37.900 --> 00:04:41.030
times 2, times 1.

00:04:41.030 --> 00:04:43.180
Now, hopefully the calculator
can handle this.

00:04:43.180 --> 00:04:47.770
And it gave us this number,
94,143,280.

00:04:47.770 --> 00:04:50.680
Let me put this on the side,
so I can read it.

00:04:50.680 --> 00:05:01.950
So this number right here
gives us 94,143,280.

00:05:01.950 --> 00:05:03.750
So that's the answer
for this problem.

00:05:03.750 --> 00:05:09.650
That there are 94,143,280
possible 9 card

00:05:09.650 --> 00:05:11.680
hands in this situation.

00:05:11.680 --> 00:05:13.840
Now, we kind of just
worked through it.

00:05:13.840 --> 00:05:15.060
We reasoned our way
through it.

00:05:15.060 --> 00:05:18.280
There is a formula for this
that does essentially the

00:05:18.280 --> 00:05:19.730
exact same thing.

00:05:19.730 --> 00:05:24.380
And the way that people denote
this formula is to say, look,

00:05:24.380 --> 00:05:32.060
we have 36 things and we are
going to choose 9 of them.

00:05:32.060 --> 00:05:32.370
Right?

00:05:32.370 --> 00:05:35.070
And we don't care about order,
so sometimes it'll be written

00:05:35.070 --> 00:05:38.440
as n choose k.

00:05:38.440 --> 00:05:39.520
Let me write it this way.

00:05:39.520 --> 00:05:40.520
So what did we do here?

00:05:40.520 --> 00:05:41.670
We have 36 things.

00:05:41.670 --> 00:05:43.010
We chose 9.

00:05:43.010 --> 00:05:47.580
So this numerator over here,
this was 36 factorial.

00:05:47.580 --> 00:05:51.970
But 36 factorial would go all
the way down to 27, 26, 25.

00:05:51.970 --> 00:05:52.810
It would just keep going.

00:05:52.810 --> 00:05:55.860
But we stopped only
nine away from 36.

00:05:55.860 --> 00:06:00.190
So this is 36 factorial, so
this part right here, that

00:06:00.190 --> 00:06:03.070
part right there, is not
just 36 factorial.

00:06:03.070 --> 00:06:09.000
It's 36 factorial divided by
36, minus 9 factorial.

00:06:09.000 --> 00:06:10.990
What is 36 minus 9?

00:06:10.990 --> 00:06:12.570
It's 27.

00:06:12.570 --> 00:06:15.580
So 27 factorial-- so let's
think about this-- 36

00:06:15.580 --> 00:06:20.540
factorial, it'd be 36 times
35, you keep going all the

00:06:20.540 --> 00:06:25.960
way, times 28 times 27, going
all the way down to 1.

00:06:25.960 --> 00:06:29.750
That is 36 factorial.

00:06:29.750 --> 00:06:31.970
Now what is 36 minus 9

00:06:31.970 --> 00:06:34.210
factorial, that's 27 factorial.

00:06:34.210 --> 00:06:39.760
So if you divide by 27
factorial, 27 factorial is 27

00:06:39.760 --> 00:06:42.370
times 26, all the
way down to 1.

00:06:42.370 --> 00:06:44.600
Well, this and this are
the exact same thing.

00:06:44.600 --> 00:06:47.820
This is 27 times 26, so that
and that would cancel out.

00:06:47.820 --> 00:06:51.960
So if you do 36 divided by 36,
minus 9 factorial, you just

00:06:51.960 --> 00:06:55.980
get the first, the largest nine
terms of 36 factorial,

00:06:55.980 --> 00:06:58.070
which is exactly what
we have over there.

00:06:58.070 --> 00:06:58.950
So that is that.

00:06:58.950 --> 00:07:00.730
And then we divided
it by 9 factorial.

00:07:04.310 --> 00:07:08.320
And this right here is
called 36 choose 9.

00:07:08.320 --> 00:07:10.270
And sometimes you'll see this
formula written like

00:07:10.270 --> 00:07:12.700
this, n choose k.

00:07:12.700 --> 00:07:18.230
And they'll write the formula as
equal to n factorial over n

00:07:18.230 --> 00:07:23.890
minus k factorial, and also in
the denominator, k factorial.

00:07:23.890 --> 00:07:27.420
And this is a general formula
that if you have n things, and

00:07:27.420 --> 00:07:31.840
you want to find out all of the
possible ways you can pick

00:07:31.840 --> 00:07:33.950
k things from those n things,
and you don't

00:07:33.950 --> 00:07:35.170
care about the order.

00:07:35.170 --> 00:07:37.390
All you care is about which k
things you picked, you don't

00:07:37.390 --> 00:07:40.100
care about the order in which
you picked those k things.

00:07:40.100 --> 00:07:42.240
So that's what we did here.


Example: Ways to arrange colors

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.580
In one game, a code made using
different colors is created by

00:00:04.580 --> 00:00:07.340
one player, the codemaker,
and the other player, the

00:00:07.340 --> 00:00:09.670
codebreaker, tries to
guess the code.

00:00:09.670 --> 00:00:12.150
The codemaker gives hints about
whether the colors are

00:00:12.150 --> 00:00:13.840
correct and in the
right position.

00:00:13.840 --> 00:00:14.890
All right.

00:00:14.890 --> 00:00:17.910
The possible colors are blue--
let me underline these in the

00:00:17.910 --> 00:00:31.340
actual colors-- blue, yellow,
white, red, orange and green.

00:00:31.340 --> 00:00:33.370
Green is already written in
green, but I'll underline it

00:00:33.370 --> 00:00:34.120
in green again.

00:00:34.120 --> 00:00:35.780
And green.

00:00:35.780 --> 00:00:38.530
How many 4-color codes
can be made if the

00:00:38.530 --> 00:00:39.970
colors cannot be repeated?

00:00:39.970 --> 00:00:42.040
To some degree, this whole
paragraph in the beginning

00:00:42.040 --> 00:00:43.590
doesn't even matter.

00:00:43.590 --> 00:00:46.120
If we're just choosing from--
let's see, we're choosing

00:00:46.120 --> 00:00:47.170
from-- how many colors
are there?

00:00:47.170 --> 00:00:52.230
There's 1, 2, 3, 4, 5, 6 colors,
and we're going to

00:00:52.230 --> 00:00:53.500
pick 4 of them.

00:00:53.500 --> 00:00:55.750
How many 4-color codes
can be made if the

00:00:55.750 --> 00:00:58.350
colors cannot be repeated?

00:00:58.350 --> 00:01:02.470
And since these are codes, we're
going to assume that

00:01:02.470 --> 00:01:15.050
blue, red, yellow and green,
that this-- that that is

00:01:15.050 --> 00:01:23.380
different than green, red,
yellow and blue.

00:01:23.380 --> 00:01:25.940
We're going to assume that these
are not the same code.

00:01:25.940 --> 00:01:28.970
Even though we've picked the
same 4 colors, we're going to

00:01:28.970 --> 00:01:31.130
assume that these are 2
different codes, and that

00:01:31.130 --> 00:01:32.410
makes sense because we're
dealing with codes.

00:01:32.410 --> 00:01:37.080
So these are different codes.

00:01:37.080 --> 00:01:39.750
So this would count as 2
different codes right here,

00:01:39.750 --> 00:01:44.120
even though we've picked
the same actual colors.

00:01:44.120 --> 00:01:45.530
The same 4 colors,
we've picked them

00:01:45.530 --> 00:01:46.310
in different orders.

00:01:46.310 --> 00:01:48.290
Now, with that out of the way,
let's think about how many

00:01:48.290 --> 00:01:50.700
different ways we can
pick 4 colors.

00:01:50.700 --> 00:01:54.530
So let's say we have
4 slots here.

00:01:54.530 --> 00:01:57.920
1 slot, 2 slot, 3 slot
and 4 slots.

00:01:57.920 --> 00:02:01.290
And at first, we care only
about, how many ways can we

00:02:01.290 --> 00:02:03.830
pick a color for that slot right
there, that first slot?

00:02:03.830 --> 00:02:05.740
We haven't picked
any colors yet.

00:02:05.740 --> 00:02:09.410
Well, we have 6 possible colors,
1, 2, 3, 4, 5, 6.

00:02:09.410 --> 00:02:12.310
So there's going to be 6
different possibilities for

00:02:12.310 --> 00:02:13.680
this slot right there.

00:02:13.680 --> 00:02:16.240
So let's put a 6 right there.

00:02:16.240 --> 00:02:19.520
Now, they told us that the
colors cannot be repeated, so

00:02:19.520 --> 00:02:21.960
whatever color is in this slot,
we're going to take it

00:02:21.960 --> 00:02:24.260
out of the possible colors.

00:02:24.260 --> 00:02:26.660
So now that we've taken that
color out, how many

00:02:26.660 --> 00:02:29.430
possibilities are when we go
to this slot, when we go to

00:02:29.430 --> 00:02:30.640
the next slot?

00:02:30.640 --> 00:02:32.030
How many possibilities
when we go to the

00:02:32.030 --> 00:02:33.240
next slot right here?

00:02:33.240 --> 00:02:36.280
Well, we took 1 of the 6 out for
the first slot, so there's

00:02:36.280 --> 00:02:38.300
only 5 possibilities here.

00:02:38.300 --> 00:02:41.140
And by the same logic when we
go to the third slot, we've

00:02:41.140 --> 00:02:44.480
used up 2 of the slots-- 2 of
the colors already, so there

00:02:44.480 --> 00:02:46.940
would only 4 possible
colors left.

00:02:46.940 --> 00:02:49.940
And then for the last slot, we
would've used up 3 of the

00:02:49.940 --> 00:02:52.840
colors, so there's only
3 possibilities left.

00:02:52.840 --> 00:02:55.350
So if we think about all of the
possibilities, all of the

00:02:55.350 --> 00:02:57.790
permutations-- and permutations
are when you

00:02:57.790 --> 00:02:59.880
think about all the
possibilities and you do care

00:02:59.880 --> 00:03:02.910
about order; where you say that
this is different than

00:03:02.910 --> 00:03:05.750
this-- this is a different
permutation than this.

00:03:05.750 --> 00:03:09.420
So all of the different
permutations here, when you

00:03:09.420 --> 00:03:14.190
pick 4 colors out of a possible
of 6 colors, it's

00:03:14.190 --> 00:03:18.680
going to be 6 possibilities for
the first 1, times 5 for

00:03:18.680 --> 00:03:24.900
the second bucket, times 4 for
the third or the third bucket

00:03:24.900 --> 00:03:27.130
of the third position,
times 3.

00:03:27.130 --> 00:03:32.950
So 6 times 5 is 30, times
4 is times 3.

00:03:32.950 --> 00:03:34.776
So 30 times 12.

00:03:34.776 --> 00:03:40.950
So this is 30 times 12, which is
equal to their 360 possible

00:03:40.950 --> 00:03:43.680
4-color codes.


Example: Ways to pick officers

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:03.610
A club of nine people wants
to choose a board of three

00:00:03.610 --> 00:00:07.940
officers: a President, a Vice
President, and a Secretary.

00:00:07.940 --> 00:00:10.370
How many ways are there
to choose the board

00:00:10.370 --> 00:00:12.430
from the nine people?

00:00:12.430 --> 00:00:15.680
Now, we're going to assume that
one person can't hold

00:00:15.680 --> 00:00:16.760
more than one office.

00:00:16.760 --> 00:00:19.910
That if I'm picked for
President, then I'm no longer

00:00:19.910 --> 00:00:22.810
a valid person for Vice
President or Secretary.

00:00:22.810 --> 00:00:25.460
So let's just think about the
three different positions.

00:00:25.460 --> 00:00:32.750
So you have the President, you
have the Vice President, VP,

00:00:32.750 --> 00:00:34.000
and then you have
the Secretary.

00:00:37.340 --> 00:00:39.840
Now, let's say that we go for
the President first. It

00:00:39.840 --> 00:00:40.950
actually doesn't matter.

00:00:40.950 --> 00:00:43.460
Let's say we were picking the
President slot first and we

00:00:43.460 --> 00:00:46.160
haven't appointed any
other slots yet.

00:00:46.160 --> 00:00:48.580
How many possibilities are
there for President?

00:00:48.580 --> 00:00:51.360
Well, the club has nine people,
so there's nine

00:00:51.360 --> 00:00:52.990
possibilities for President.

00:00:52.990 --> 00:00:54.950
There's going to be nine
possibilities for President.

00:00:54.950 --> 00:00:58.090
Now, we're going to pick
one of those nine.

00:00:58.090 --> 00:01:00.200
We're going to kind of take them
out of the running for

00:01:00.200 --> 00:01:01.590
the other two offices.

00:01:01.590 --> 00:01:01.900
Right?

00:01:01.900 --> 00:01:03.890
Because someone's going to be
President, so one of the nine

00:01:03.890 --> 00:01:04.650
is going to be President.

00:01:04.650 --> 00:01:07.300
There's nine possibilities, but
one of the nine is going

00:01:07.300 --> 00:01:08.260
to be President.

00:01:08.260 --> 00:01:10.130
So you take that person aside.

00:01:10.130 --> 00:01:11.750
He or she is now
the President.

00:01:11.750 --> 00:01:14.000
How many people are left
to be Vice President?

00:01:14.000 --> 00:01:15.190
Well, now there's only
eight possible

00:01:15.190 --> 00:01:16.510
candidates for Vice President?

00:01:16.510 --> 00:01:17.900
Eight possibilities.

00:01:17.900 --> 00:01:19.990
Now he or she also goes aside.

00:01:19.990 --> 00:01:22.390
Now how many people are
left for Secretary?

00:01:22.390 --> 00:01:26.210
Well, now there's only seven
possibilities for Secretary.

00:01:26.210 --> 00:01:28.720
So if you want to think about
all of the different ways

00:01:28.720 --> 00:01:33.170
there are to choose a board from
the nine people, there's

00:01:33.170 --> 00:01:38.550
the 9 for President times the 8
for Vice President times the

00:01:38.550 --> 00:01:41.510
7 for Secretary.

00:01:41.510 --> 00:01:42.450
You didn't have to
do it this way.

00:01:42.450 --> 00:01:43.990
It could've been-- you could've
picked Secretary

00:01:43.990 --> 00:01:45.890
first. Then there would've
been nine choices.

00:01:45.890 --> 00:01:47.100
And then you could've picked
Vice President, and there

00:01:47.100 --> 00:01:48.100
would've still been
eight choices.

00:01:48.100 --> 00:01:49.680
And then you could've picked
President last, and there

00:01:49.680 --> 00:01:50.800
would've only been
seven choices.

00:01:50.800 --> 00:01:53.430
But either way you would've
got 9 times 8 times 7.

00:01:53.430 --> 00:02:04.400
And that is, let's see, 9 times
8 is 72 times 7 is-- 2

00:02:04.400 --> 00:02:10.130
times 7 is 14, 7 times
7 is 49 plus 1 is 50.

00:02:10.130 --> 00:02:15.670
So 504 possible ways to pick
your board out of a club of

00:02:15.670 --> 00:02:17.790
only nine people.


Die rolling probability with independent events

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.750
Find the probability of rolling
even numbers three

00:00:03.750 --> 00:00:08.740
times using a six-sided die
numbered from 1 to 6.

00:00:08.740 --> 00:00:10.860
So let's just figure out the
probability of rolling it each

00:00:10.860 --> 00:00:11.490
of the times.

00:00:11.490 --> 00:00:15.020
So the probability of rolling
even numbers.

00:00:15.020 --> 00:00:22.690
So even roll on six-sided die.

00:00:22.690 --> 00:00:24.310
So let's think about
that probability.

00:00:24.310 --> 00:00:26.360
Well, how many total
outcomes are there?

00:00:26.360 --> 00:00:28.180
How many possible rolls
could we get?

00:00:28.180 --> 00:00:32.119
Well, you get one, two, three,
four, five, six.

00:00:32.119 --> 00:00:34.890
And how many of them satisfy
these conditions, that it's an

00:00:34.890 --> 00:00:36.420
even number?

00:00:36.420 --> 00:00:38.990
Well, it could be a 2,
it could be a 4, or

00:00:38.990 --> 00:00:40.440
it could be a 6.

00:00:40.440 --> 00:00:44.160
So the probability is the events
that match what you

00:00:44.160 --> 00:00:47.830
need, your condition for right
here, so three of the possible

00:00:47.830 --> 00:00:49.680
events are an even roll.

00:00:49.680 --> 00:00:55.310
And it's out of a total of
six possible events.

00:00:55.310 --> 00:00:58.700
So there is a-- 3 over 6 is
the same thing as 1/2

00:00:58.700 --> 00:01:02.070
probability of rolling
even on each roll.

00:01:02.070 --> 00:01:03.770
Now they're going to
roll-- they want to

00:01:03.770 --> 00:01:05.860
roll even three times.

00:01:05.860 --> 00:01:07.770
And these are all going to
be independent events.

00:01:07.770 --> 00:01:10.260
Every time you roll, it's not
going to affect what happens

00:01:10.260 --> 00:01:13.380
in the next roll, despite what
some gamblers might think.

00:01:13.380 --> 00:01:16.730
It has no impact on what happens
on the next roll.

00:01:16.730 --> 00:01:30.160
So the probability of rolling
even three times is equal to

00:01:30.160 --> 00:01:38.670
the probability of an even roll
one time, or even roll on

00:01:38.670 --> 00:01:44.410
six-sided die-- this thing over
here is equal to that

00:01:44.410 --> 00:01:49.210
thing times that thing again.

00:01:49.210 --> 00:01:53.510
All right, that's our first
roll-- we copy and we paste

00:01:53.510 --> 00:02:01.090
it-- times that thing and then
times that thing again.

00:02:01.090 --> 00:02:01.410
Right?

00:02:01.410 --> 00:02:03.560
That's our first roll,
which is that.

00:02:03.560 --> 00:02:04.470
That's our second roll.

00:02:04.470 --> 00:02:05.210
That's our third roll.

00:02:05.210 --> 00:02:06.540
They're independent events.

00:02:06.540 --> 00:02:10.150
So this is going to be equal to
1/2-- that's the same 1/2

00:02:10.150 --> 00:02:14.560
right there-- times 1/2
times 1/2, which is

00:02:14.560 --> 00:02:16.240
equal to 1 over 8.

00:02:16.240 --> 00:02:23.080
There's a 1 in 8 possibility
that you roll even numbers on

00:02:23.080 --> 00:02:24.620
all three rolls.

00:02:24.620 --> 00:02:27.420
On this roll, this roll,
and that roll.


Test taking probability and independent events

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:05.040
On a multiple choice test,
problem 1 has 4 choices, and

00:00:05.040 --> 00:00:07.490
problem 2 has 3 choices.

00:00:07.490 --> 00:00:08.910
That should be choices.

00:00:08.910 --> 00:00:11.770
Each problem has only
one correct answer.

00:00:11.770 --> 00:00:14.830
What is the probability of
randomly guessing the correct

00:00:14.830 --> 00:00:16.700
answer on both problems?

00:00:16.700 --> 00:00:20.100
Now, the probability of guessing
the correct answer on

00:00:20.100 --> 00:00:22.685
each problem-- these are
independent events.

00:00:22.685 --> 00:00:23.830
So let's write this down.

00:00:23.830 --> 00:00:34.380
The probability of correct
on problem number 1 is

00:00:34.380 --> 00:00:35.160
independent.

00:00:35.160 --> 00:00:36.300
Or let me write it this way.

00:00:36.300 --> 00:00:42.090
Probability of correct on number
1 and probability of

00:00:42.090 --> 00:00:49.136
correct on number 2, on problem
2, are independent.

00:00:53.510 --> 00:00:56.650
Which means that the outcome
of one of the events, of

00:00:56.650 --> 00:01:00.650
guessing on the first problem,
isn't going to affect the

00:01:00.650 --> 00:01:01.950
probability of guessing
correctly

00:01:01.950 --> 00:01:04.129
on the second problem.

00:01:04.129 --> 00:01:09.250
Independent events.

00:01:09.250 --> 00:01:13.800
So the probability of guessing
on both of them-- so that

00:01:13.800 --> 00:01:22.030
means that the probability of
being correct-- on guessing

00:01:22.030 --> 00:01:27.320
correct on 1 and number 2 is
going to be equal to the

00:01:27.320 --> 00:01:29.270
product of these
probabilities.

00:01:29.270 --> 00:01:31.480
And we're going to see why that
is visually in a second.

00:01:31.480 --> 00:01:35.620
But it's going to be the
probability of correct on

00:01:35.620 --> 00:01:43.990
number 1 times the probability
of being correct on number 2.

00:01:43.990 --> 00:01:46.310
Now, what are each of
these probabilities?

00:01:46.310 --> 00:01:50.170
On number 1, there are 4
choices, there are 4 possible

00:01:50.170 --> 00:01:53.620
outcomes, and only one of them
is going to be correct.

00:01:53.620 --> 00:01:56.040
Each one only has one
correct answer.

00:01:56.040 --> 00:01:59.640
So the probability of being
correct on problem 1 is 1/4.

00:01:59.640 --> 00:02:01.480
And then the probability of
being correct on problem

00:02:01.480 --> 00:02:05.010
number 2-- problem number 2 has
three choices, so there's

00:02:05.010 --> 00:02:06.960
three possible outcomes.

00:02:06.960 --> 00:02:08.810
And there's only one correct
one, so only

00:02:08.810 --> 00:02:09.949
one of them are correct.

00:02:09.949 --> 00:02:12.920
So probability of correct
on number 2 is 1/3.

00:02:12.920 --> 00:02:15.370
Probability of guessing correct
on number 1 is 1/4.

00:02:15.370 --> 00:02:19.020
The probability of doing on both
of them is going to be

00:02:19.020 --> 00:02:20.010
its product.

00:02:20.010 --> 00:02:26.450
So it's going to be equal to
1/4 times 1/3 is 1/12.

00:02:26.450 --> 00:02:30.305
Now, to see kind of visually
why this make sense, let's

00:02:30.305 --> 00:02:31.370
draw a little chart here.

00:02:31.370 --> 00:02:35.850
And we did a similar thing for
when we thought about rolling

00:02:35.850 --> 00:02:37.780
two separate dice.

00:02:37.780 --> 00:02:40.330
So let's think about
problem number 1.

00:02:40.330 --> 00:02:43.750
Problem number 1 has
4 choices, only

00:02:43.750 --> 00:02:45.480
one of which is correct.

00:02:45.480 --> 00:02:47.950
So let's write-- so
it has 4 choices.

00:02:47.950 --> 00:02:51.810
So it has 1-- let's write
incorrect choice 1, incorrect

00:02:51.810 --> 00:02:56.120
choice 2, incorrect choice 3,
and then it has the correct

00:02:56.120 --> 00:02:57.480
choice over there.

00:02:57.480 --> 00:02:58.620
So those are the 4 choices.

00:02:58.620 --> 00:03:00.180
They're not going to necessarily
be in that order

00:03:00.180 --> 00:03:02.740
on the exam, but we can just
list them in this order.

00:03:02.740 --> 00:03:10.160
Now problem number 2
has 3 choices, only

00:03:10.160 --> 00:03:11.900
one of which is correct.

00:03:11.900 --> 00:03:15.700
So problem number 2 has
incorrect choice 1, incorrect

00:03:15.700 --> 00:03:18.850
choice 2, and then let's say the
third choice is correct.

00:03:18.850 --> 00:03:20.800
It's not necessarily in that
order, but we know it has 2

00:03:20.800 --> 00:03:22.760
incorrect and 1 correct
choices.

00:03:22.760 --> 00:03:25.970
Now, what are all of the
different possible outcomes?

00:03:25.970 --> 00:03:27.550
We can draw a little
bit of a grid here.

00:03:30.570 --> 00:03:34.550
All of these possible
outcomes.

00:03:34.550 --> 00:03:36.220
Let's draw all of
the outcomes.

00:03:36.220 --> 00:03:39.630
Each of these cells or each of
these boxes in a grid are a

00:03:39.630 --> 00:03:40.900
possible outcome.

00:03:40.900 --> 00:03:42.290
You could-- you're
just guessing.

00:03:42.290 --> 00:03:44.350
You're randomly choosing one
of these 4, you're randomly

00:03:44.350 --> 00:03:45.850
choosing one of these 4.

00:03:45.850 --> 00:03:49.000
So you might get incorrect
choice 1 and incorrect choice

00:03:49.000 --> 00:03:51.690
1-- incorrect choice in problem
number 1 and then

00:03:51.690 --> 00:03:53.020
incorrect choice in
problem number 2.

00:03:53.020 --> 00:03:54.870
That would be that
cell right there.

00:03:54.870 --> 00:03:58.150
Maybe you get this-- maybe you
get problem number 1 correct,

00:03:58.150 --> 00:04:01.650
but you get incorrect choice
number 2 in problem number 2.

00:04:01.650 --> 00:04:03.880
So these would represent all of
the possible outcomes when

00:04:03.880 --> 00:04:05.200
you guess on each problem.

00:04:05.200 --> 00:04:07.700
And which of these outcomes
represent

00:04:07.700 --> 00:04:09.570
getting correct on both?

00:04:09.570 --> 00:04:11.800
Well, getting correct on both
is only this one, correct on

00:04:11.800 --> 00:04:16.190
choice 1 and correct on choice--
on problem number 2.

00:04:16.190 --> 00:04:19.430
And so that's one of the
possible outcomes and how many

00:04:19.430 --> 00:04:20.510
total outcomes are there?

00:04:20.510 --> 00:04:25.930
There's 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, out of 12

00:04:25.930 --> 00:04:26.870
possible outcomes.

00:04:26.870 --> 00:04:29.960
Or since these are independent
events, you can multiply.

00:04:29.960 --> 00:04:32.910
You see that they're 12 outcomes
because there's 12

00:04:32.910 --> 00:04:34.170
possible outcomes.

00:04:34.170 --> 00:04:37.340
So there's 4 possible outcomes
for problem number 1, times

00:04:37.340 --> 00:04:40.050
the 3 possible outcomes for
problem number 2, and that's

00:04:40.050 --> 00:04:42.330
also where you get a 12.


Independent or dependent probability event?

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.190
The marching band is holding a
raffle at a football game with

00:00:04.190 --> 00:00:05.340
two prizes.

00:00:05.340 --> 00:00:07.790
After the first ticket is pulled
out and the winner

00:00:07.790 --> 00:00:11.100
determined, the ticket is
taped to the prize.

00:00:11.100 --> 00:00:14.070
The next ticket is pulled out to
determine the winner of the

00:00:14.070 --> 00:00:15.560
second prize.

00:00:15.560 --> 00:00:18.660
Are the two events
independent?

00:00:18.660 --> 00:00:19.890
Explain.

00:00:19.890 --> 00:00:22.480
Now before we even think about
this exact case, let's think

00:00:22.480 --> 00:00:25.040
about what it means for events
to be independent.

00:00:25.040 --> 00:00:28.980
It means that the outcome of one
event doesn't affect the

00:00:28.980 --> 00:00:31.220
outcome of the other event.

00:00:31.220 --> 00:00:35.040
Now in this situation, the first
event-- after the first

00:00:35.040 --> 00:00:37.650
ticket is pulled out and the
winner determined-- the ticket

00:00:37.650 --> 00:00:41.410
is taped to the prize.

00:00:41.410 --> 00:00:43.990
Then the next ticket is pulled
out to determine the winner of

00:00:43.990 --> 00:00:45.360
the second prize.

00:00:45.360 --> 00:00:48.490
Now, the winner of the second
prize-- the possible winners,

00:00:48.490 --> 00:00:53.220
the possible outcomes for the
second prize, is dependent on

00:00:53.220 --> 00:00:55.890
who was pulled out for
the first prize.

00:00:55.890 --> 00:01:00.300
You can imagine if there's
three tickets, let's say

00:01:00.300 --> 00:01:05.510
there's tickets A, B,
and C in the bag.

00:01:05.510 --> 00:01:12.020
And for the first prize,
they pull out ticket A.

00:01:12.020 --> 00:01:15.430
That's for the first prize.

00:01:15.430 --> 00:01:18.110
Now, when we think about who
could be pulled out for the

00:01:18.110 --> 00:01:24.000
second prize, it's only going
to be tickets B or C.

00:01:24.000 --> 00:01:25.800
Now the first prize could
have gone the other way.

00:01:25.800 --> 00:01:29.260
It could have been
A, B, and C.

00:01:29.260 --> 00:01:33.120
The first prize could have
gone to ticket B.

00:01:33.120 --> 00:01:36.120
And then the possible outcomes
for the second prize

00:01:36.120 --> 00:01:39.940
would be A or C.

00:01:39.940 --> 00:01:43.390
So the possible outcomes for
the second event, for the

00:01:43.390 --> 00:01:49.500
second prize, are completely
dependent on what happened or

00:01:49.500 --> 00:01:52.310
what ticket was pulled out
for the first prize.

00:01:52.310 --> 00:01:54.440
So these are not independent
events.

00:01:54.440 --> 00:01:57.500
The second event-- the outcomes
for it, are dependent

00:01:57.500 --> 00:01:59.180
on what happened in
the first event.

00:01:59.180 --> 00:02:03.450
So they are not independent.

00:02:06.150 --> 00:02:08.949
The way that we could have
made them independent is,

00:02:08.949 --> 00:02:12.330
after the first ticket was
pulled out, if they just wrote

00:02:12.330 --> 00:02:14.170
down the name or something,
and then put

00:02:14.170 --> 00:02:15.660
that ticket back in.

00:02:15.660 --> 00:02:16.780
Instead they taped
it to the prize.

00:02:16.780 --> 00:02:19.550
But if they put that ticket
back in, then the second

00:02:19.550 --> 00:02:22.260
prize, it would have still had
all the tickets there.

00:02:22.260 --> 00:02:25.300
It wouldn't have mattered who
was picked out in the first

00:02:25.300 --> 00:02:26.980
time because their name was just
written down, but their

00:02:26.980 --> 00:02:28.210
ticket was put back in.

00:02:28.210 --> 00:02:30.290
And then you would have
been independent.

00:02:30.290 --> 00:02:31.980
So if you had replaced the
ticket, you would have been

00:02:31.980 --> 00:02:32.720
independent.

00:02:32.720 --> 00:02:34.748
But since they didn't replace
the ticket, they taped it to

00:02:34.748 --> 00:02:38.510
the prize, these are not
independent events.


Example: Probability through counting outcomes

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.820
Find the probability
of flipping exactly

00:00:02.820 --> 00:00:04.900
two heads on 3 coins.

00:00:04.900 --> 00:00:07.340
So to figure out this
probability, a good place to

00:00:07.340 --> 00:00:10.430
start is just to think about all
of the different possible

00:00:10.430 --> 00:00:13.180
ways that we can flip 3 coins.

00:00:13.180 --> 00:00:16.370
So we could get all tails.

00:00:16.370 --> 00:00:18.770
Tails, tails, tails.

00:00:18.770 --> 00:00:22.020
We could get tails,
tails, heads.

00:00:22.020 --> 00:00:25.280
We could get tails,
heads, tails.

00:00:25.280 --> 00:00:28.760
We could get tails,
heads, heads.

00:00:28.760 --> 00:00:32.340
We could get heads,
tails, tails.

00:00:32.340 --> 00:00:35.960
We could get heads,
tails, heads.

00:00:35.960 --> 00:00:39.900
We could get heads,
heads, tails.

00:00:39.900 --> 00:00:41.730
And then we could
get all heads.

00:00:41.730 --> 00:00:43.850
We could get all heads
over here.

00:00:43.850 --> 00:00:49.380
So there are 1, 2, 3, 4, 5,
6, 7, 8 possible outcomes.

00:00:49.380 --> 00:00:55.000
8 possible outcomes.

00:00:55.000 --> 00:00:56.960
Now how many of the
outcomes involve

00:00:56.960 --> 00:01:01.180
flipping exactly 2 heads?

00:01:01.180 --> 00:01:02.260
Let's see, that's all tails.

00:01:02.260 --> 00:01:03.300
That's 1 head, 1 head.

00:01:03.300 --> 00:01:05.920
This has 2 heads right there.

00:01:05.920 --> 00:01:06.670
That's 1 head.

00:01:06.670 --> 00:01:09.210
This is 2 heads right
over there.

00:01:09.210 --> 00:01:11.840
Then this is 2 heads
right over here.

00:01:11.840 --> 00:01:14.270
And then this is 3 heads,
so that doesn't count.

00:01:14.270 --> 00:01:30.010
So there are 3 outcomes
with exactly 2 heads.

00:01:30.010 --> 00:01:32.490
So, let me spell
heads properly.

00:01:32.490 --> 00:01:34.730
2 heads.

00:01:34.730 --> 00:01:44.690
So the probability of flipping
exactly 2 heads-- And the word

00:01:44.690 --> 00:01:47.410
exactly is important, because
if you didn't say exactly,

00:01:47.410 --> 00:01:49.895
then maybe 3 heads, when you
flip 2 heads, so we have to

00:01:49.895 --> 00:01:50.920
say exactly 2 heads.

00:01:50.920 --> 00:01:54.090
So you don't include
the situation

00:01:54.090 --> 00:01:55.220
where you get 3 heads.

00:01:55.220 --> 00:01:59.330
So the probability of flipping
exactly 2 heads is equal to

00:01:59.330 --> 00:02:01.700
the 3 outcomes with 2 heads
divided by the 8 possible

00:02:01.700 --> 00:02:03.670
outcomes, or 3/8.

00:02:03.670 --> 00:02:06.340
So it is equal to 3/8.

00:02:06.340 --> 00:02:08.470
And we are done.


Events and Outcomes 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.210
Find the probability of
rolling doubles on two

00:00:04.210 --> 00:00:08.890
six-sided dice, numbered
from 1 to 6.

00:00:08.890 --> 00:00:10.100
So let's just think about it.

00:00:10.100 --> 00:00:12.720
Let's think about all of the
outcomes, all of the possible

00:00:12.720 --> 00:00:15.030
outcomes, and then what fraction
of those outcomes

00:00:15.030 --> 00:00:17.670
represent us getting doubles.

00:00:17.670 --> 00:00:20.380
We want to roll doubles.

00:00:20.380 --> 00:00:22.640
So let's make a little table
here that'll give us all of

00:00:22.640 --> 00:00:23.740
our outcomes.

00:00:23.740 --> 00:00:27.450
You have 6 outcomes on the first
dice, or the first die I

00:00:27.450 --> 00:00:27.900
should say.

00:00:27.900 --> 00:00:34.950
So 1, 2, 3, 4, 5, 6.

00:00:34.950 --> 00:00:36.770
So let's call this
the first die.

00:00:40.060 --> 00:00:43.270
And then the second die,
there are 6 outcomes.

00:00:43.270 --> 00:00:48.550
So 1, 2, 3, 4, 5, 6.

00:00:48.550 --> 00:00:50.115
This is the second die.

00:00:52.750 --> 00:00:55.230
So all of the different possible
outcomes are going to

00:00:55.230 --> 00:00:57.760
be some-- you're going to get an
outcome-- so you could get

00:00:57.760 --> 00:00:58.710
a 3 over here.

00:00:58.710 --> 00:01:00.550
And then when you get a 3 on the
second die, you might have

00:01:00.550 --> 00:01:03.260
gotten a 1, 2, 3, 4, 5,
or 6 on the first die.

00:01:03.260 --> 00:01:05.800
Or you might have gotten a 5 on
the first die and then you

00:01:05.800 --> 00:01:08.560
could get a 1, 2 3, 4, 5,
or 6 on the second die.

00:01:08.560 --> 00:01:10.580
So if I draw a grid here.

00:01:10.580 --> 00:01:13.930
If I were to draw a grid here,
let me do my best to draw a

00:01:13.930 --> 00:01:16.456
quick grid.

00:01:16.456 --> 00:01:20.310
Just draw it really fast. So
let me just draw this grid

00:01:20.310 --> 00:01:22.520
right here.

00:01:22.520 --> 00:01:26.280
Each cell of this grid, each
box in this grid, would

00:01:26.280 --> 00:01:27.530
represent an outcome.

00:01:31.610 --> 00:01:33.990
All of the possible outcomes.

00:01:33.990 --> 00:01:36.520
Let me finish drawing it.

00:01:36.520 --> 00:01:37.540
So there you go.

00:01:37.540 --> 00:01:40.000
And so just to get an
understanding of what I mean

00:01:40.000 --> 00:01:42.140
by it represents an outcome.

00:01:42.140 --> 00:01:47.280
This cell right here would
represent getting a 4 on your

00:01:47.280 --> 00:01:50.750
first die and a 2 on
your second die.

00:01:50.750 --> 00:01:52.820
That's the outcome
it represents.

00:01:52.820 --> 00:01:54.680
It's one of the potential
outcomes.

00:01:54.680 --> 00:01:56.590
Now how many total outcomes
are there?

00:01:56.590 --> 00:01:58.850
Well you could look at the
area of this right here.

00:01:58.850 --> 00:02:01.470
The area of this is going
to be 6 times 6.

00:02:01.470 --> 00:02:04.080
You have 36 possible outcomes.

00:02:04.080 --> 00:02:10.639
So 36 possible outcomes.

00:02:10.639 --> 00:02:12.260
And it makes sense.

00:02:12.260 --> 00:02:13.480
You can think about
the first die.

00:02:13.480 --> 00:02:14.970
There's 6 outcomes.

00:02:14.970 --> 00:02:18.380
And then for each of those 6
outcomes, you have 6 possible

00:02:18.380 --> 00:02:19.830
outcomes for the second die.

00:02:19.830 --> 00:02:21.560
So 6 times 6.

00:02:21.560 --> 00:02:23.450
Or you can think of in
this grid, it has

00:02:23.450 --> 00:02:25.850
exactly 36 cells in it.

00:02:25.850 --> 00:02:29.640
Now what are the outcomes that
satisfy our conditions?

00:02:29.640 --> 00:02:33.900
That entail us getting
doubles?

00:02:33.900 --> 00:02:35.330
What are the outcomes there?

00:02:35.330 --> 00:02:37.260
Well we could get a 1.

00:02:37.260 --> 00:02:39.100
If we get a 1 on the first die,
we have to get a 1 on the

00:02:39.100 --> 00:02:39.810
second die.

00:02:39.810 --> 00:02:42.150
If we get a 2 on the first die,
we have to get a 2 on the

00:02:42.150 --> 00:02:42.880
second die.

00:02:42.880 --> 00:02:45.190
If we get a 3 on the first die,
we have to get a 3 on the

00:02:45.190 --> 00:02:46.180
second die.

00:02:46.180 --> 00:02:47.360
I think you see where
this is going.

00:02:47.360 --> 00:02:49.710
4 and 4.

00:02:49.710 --> 00:02:51.940
5 and 5.

00:02:51.940 --> 00:02:53.720
And then 6 and 6.

00:02:53.720 --> 00:02:57.420
So how many of these outcomes
involve us getting double?

00:02:57.420 --> 00:03:02.990
Well, it's 1, 2, 3,
4, 5, 6 outcomes.

00:03:02.990 --> 00:03:11.420
So there's 6 outcomes
with doubles.

00:03:11.420 --> 00:03:12.690
So if someone asks
you, what's the

00:03:12.690 --> 00:03:17.990
probability of getting doubles?

00:03:17.990 --> 00:03:21.250
The probability of getting
doubles when you rolled two

00:03:21.250 --> 00:03:24.230
six-sided dice, numbered
from 1 to 6.

00:03:24.230 --> 00:03:28.390
It's going to be equal to the
number of outcomes that give

00:03:28.390 --> 00:03:31.830
us doubles-- so it's going to
be equal to 6-- over the

00:03:31.830 --> 00:03:34.580
number of the total number
of possible outcomes.

00:03:34.580 --> 00:03:36.520
Over 36.

00:03:36.520 --> 00:03:40.360
And 6 over 36 is 1/6.

00:03:40.360 --> 00:03:43.670
Now I made sure that you kind of
understood that look, these

00:03:43.670 --> 00:03:45.320
are all the possible outcomes.

00:03:45.320 --> 00:03:47.130
And 6 of them satisfy.

00:03:47.130 --> 00:03:48.240
So 6 of the 36.

00:03:48.240 --> 00:03:51.180
So there's a 1/6 probability
of rolling doubles.

00:03:51.180 --> 00:03:53.390
But another way you could have
thought about it is, you know

00:03:53.390 --> 00:03:55.000
what, let's just roll
the first die.

00:03:55.000 --> 00:03:56.730
It's going to have some
number on it, right?

00:03:56.730 --> 00:03:58.610
The first die is going to
have some number on it.

00:03:58.610 --> 00:04:01.630
Now in order for me to have
doubles, the second die has to

00:04:01.630 --> 00:04:04.010
have that same exact number
on it, right?

00:04:04.010 --> 00:04:05.410
So I don't care what
the first die is.

00:04:05.410 --> 00:04:07.950
It could be a 3, it could
be a 4, it could be a 5.

00:04:07.950 --> 00:04:08.960
I don't know what that is.

00:04:08.960 --> 00:04:11.400
But after I've rolled that first
die, in order to get

00:04:11.400 --> 00:04:14.630
doubles, what the second die
roll is-- they might be almost

00:04:14.630 --> 00:04:18.200
simultaneous-- but what the
second die rolls matters.

00:04:18.200 --> 00:04:21.890
And so whatever number I have
on the first die, there's a

00:04:21.890 --> 00:04:27.070
1/6 chance that the second die
is going to have that same

00:04:27.070 --> 00:04:27.770
number on it.

00:04:27.770 --> 00:04:29.430
So that's another way
of reasoning that

00:04:29.430 --> 00:04:31.260
you have a 1/6 chance.

00:04:31.260 --> 00:04:34.180
The first die does anything it
wants, but there's only a 1 in

00:04:34.180 --> 00:04:36.410
6 chance that the second die
is going to have the same

00:04:36.410 --> 00:04:38.420
thing as the first die.

00:04:38.420 --> 00:04:41.050
Either way, you get to
the same 1/6 answer.


Deductive reasoning 3

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.630
Use deductive reasoning and the
distributive property to

00:00:03.630 --> 00:00:08.360
justify x plus y squared is
equal to x squared plus 2xy

00:00:08.360 --> 00:00:09.690
plus y squared.

00:00:09.690 --> 00:00:12.190
Provide the reasoning
for each step.

00:00:12.190 --> 00:00:14.290
Now when they say use deductive
reasoning and all

00:00:14.290 --> 00:00:16.920
this stuff, it might seem like
something daunting and new,

00:00:16.920 --> 00:00:18.940
but this is no different than
what we've done in the past.

00:00:18.940 --> 00:00:20.970
In fact, we've done this
very exact problem.

00:00:20.970 --> 00:00:23.700
So let's just do it step by
step, show our logic, and

00:00:23.700 --> 00:00:25.020
that's essentially deductive
reasoning.

00:00:25.020 --> 00:00:27.800
We're starting with a statement
and we're going to

00:00:27.800 --> 00:00:30.330
deduce that this is going to
be equal to something else.

00:00:30.330 --> 00:00:32.379
So let's start with
x plus y squared.

00:00:39.090 --> 00:00:42.550
We know an exponent means to
multiply something by itself

00:00:42.550 --> 00:00:43.590
that many times.

00:00:43.590 --> 00:00:47.060
So we know that this is the same
thing, or we can deduce

00:00:47.060 --> 00:00:51.130
that it is the same thing, as
x plus y times-- and I'll do

00:00:51.130 --> 00:00:53.650
this next x plus y in
a different color--

00:00:53.650 --> 00:00:57.100
times x plus y.

00:00:57.100 --> 00:00:59.390
That's what x plus
y squared is.

00:00:59.390 --> 00:01:01.630
Now they say to use the
distributive property.

00:01:01.630 --> 00:01:02.970
And this is a bit
of review here.

00:01:02.970 --> 00:01:04.950
And we've seen it many, many,
many times before.

00:01:04.950 --> 00:01:08.580
The distributive property just
tells us that if we have a

00:01:08.580 --> 00:01:16.100
times b plus c-- I want to do
as many colors as possible--

00:01:16.100 --> 00:01:26.270
that this is equal to a times
each of these terms. a times b

00:01:26.270 --> 00:01:28.340
plus a times c.

00:01:31.912 --> 00:01:33.320
It's called distributive
property cause you're

00:01:33.320 --> 00:01:36.710
distributing the a in all of
the terms in the expression

00:01:36.710 --> 00:01:39.180
that your multiplying a by.

00:01:39.180 --> 00:01:41.710
Now we can do the exact
same thing here.

00:01:41.710 --> 00:01:46.030
Instead of an a, you could
imagine this is an x plus y.

00:01:46.030 --> 00:01:48.880
And we can take this entire x
plus y and we can distribute

00:01:48.880 --> 00:01:52.470
it on to both of the terms on
this expression that it is

00:01:52.470 --> 00:01:53.000
multiplying.

00:01:53.000 --> 00:01:55.450
If this was an a, it'd
be ax plus ay.

00:01:55.450 --> 00:01:58.570
Now that it's an x plus y, we
multiply the x plus y times

00:01:58.570 --> 00:02:00.640
each of those terms. And
that's just by the

00:02:00.640 --> 00:02:02.550
distributive property.

00:02:02.550 --> 00:02:04.620
So by the distributive property
that's going to be

00:02:04.620 --> 00:02:08.264
equal to-- we'll distribute this
on to each of them-- x

00:02:08.264 --> 00:02:09.514
plus y times x.

00:02:12.975 --> 00:02:15.085
And actually I don't even have
to write the x after it.

00:02:15.085 --> 00:02:15.940
I could just write it there.

00:02:15.940 --> 00:02:19.200
It doesn't matter whether you
multiply x times x plus y, or

00:02:19.200 --> 00:02:20.520
x plus y times x.

00:02:20.520 --> 00:02:21.350
Order doesn't matter.

00:02:21.350 --> 00:02:22.860
So that's that times that.

00:02:22.860 --> 00:02:26.960
And then it's going to be
plus y times x plus y.

00:02:31.530 --> 00:02:34.860
And now we can apply the
distributive property again.

00:02:34.860 --> 00:02:37.565
We have x being multiplied by
x plus y, then we have a y

00:02:37.565 --> 00:02:39.570
being multiplied by x plus y.

00:02:39.570 --> 00:02:41.510
So let's just do that again.

00:02:41.510 --> 00:02:51.245
So then we get this is equal to
x times x plus x times y.

00:02:54.410 --> 00:02:57.270
I'm going through great pains to
keep the colors consistent.

00:02:57.270 --> 00:03:00.280
Plus y times x.

00:03:00.280 --> 00:03:04.970
Plus y times this x over here.

00:03:04.970 --> 00:03:10.930
Plus y times that
y over there.

00:03:10.930 --> 00:03:12.450
I'm doing this a lot
slower, and I'm not

00:03:12.450 --> 00:03:13.600
skipping any steps here.

00:03:13.600 --> 00:03:16.770
Now what do each of these
things equal? x times x.

00:03:16.770 --> 00:03:18.510
That is the same thing
as x squared.

00:03:18.510 --> 00:03:20.850
So this is equal to x squared.

00:03:20.850 --> 00:03:23.450
This right here, xy,
we have one xy.

00:03:23.450 --> 00:03:27.220
But then we have yx is also
the same thing as xy.

00:03:27.220 --> 00:03:29.270
It doesn't matter what order
you multiply it in.

00:03:29.270 --> 00:03:32.920
So xy plus xy is 2xy.

00:03:32.920 --> 00:03:34.580
Plus 2xy.

00:03:34.580 --> 00:03:37.750
And then this last term right
here, y times y, that's the

00:03:37.750 --> 00:03:40.310
same thing as y squared.

00:03:40.310 --> 00:03:41.330
So we're done.

00:03:41.330 --> 00:03:43.130
We've used deductive
reasoning.

00:03:43.130 --> 00:03:47.110
We've just used logical steps to
start with a statement, to

00:03:47.110 --> 00:03:48.930
start with an expression
really.

00:03:48.930 --> 00:03:52.040
And we essentially just
logically manipulated it.

00:03:52.040 --> 00:03:53.440
We started with.

00:03:53.440 --> 00:03:56.210
I guess you could call this a
statement-- I guess that's the

00:03:56.210 --> 00:03:58.790
best thing to call it-- and we
logically manipulated to come

00:03:58.790 --> 00:04:02.250
up with another statement,
another fact.

00:04:02.250 --> 00:04:06.290
We know that this is equal to
this using logical properties

00:04:06.290 --> 00:04:08.090
and distributive property
and things like that and

00:04:08.090 --> 00:04:09.340
properties of exponents.


Finding probability example 3

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.850
Find the probability of pulling
a yellow marble from a

00:00:03.850 --> 00:00:09.060
bag with three yellow, two red,
two green, and one blue

00:00:09.060 --> 00:00:10.480
marbles, I'm assuming.

00:00:10.480 --> 00:00:12.510
So let's just think about
this bag a little bit.

00:00:12.510 --> 00:00:14.470
There are three yellow
marbles in it.

00:00:14.470 --> 00:00:15.660
So let me draw them.

00:00:15.660 --> 00:00:19.350
We have one, two, three
yellow marbles.

00:00:19.350 --> 00:00:21.010
We have two red marbles.

00:00:23.565 --> 00:00:27.170
So we have one, two red marbles,
assuming they're all

00:00:27.170 --> 00:00:28.240
mixed in there.

00:00:28.240 --> 00:00:32.990
We have two green marbles--
one, two green marbles.

00:00:32.990 --> 00:00:34.500
And then there's one
blue marble.

00:00:37.720 --> 00:00:39.820
And they want to know the
probability of pulling a

00:00:39.820 --> 00:00:40.700
yellow marble.

00:00:40.700 --> 00:00:48.530
So they want to know what is the
probability of pulling a

00:00:48.530 --> 00:00:53.430
yellow marble.

00:00:53.430 --> 00:00:57.470
Now probability is just a way of
expressing-- well, I don't

00:00:57.470 --> 00:00:59.330
want to use too loaded words.

00:00:59.330 --> 00:01:02.460
But it's the odds of picking
yours relative to the odds of

00:01:02.460 --> 00:01:03.440
picking everything else.

00:01:03.440 --> 00:01:06.130
Or another way of thinking about
it is, it's the number

00:01:06.130 --> 00:01:09.330
of outcomes that could result
in what you want, a yellow

00:01:09.330 --> 00:01:13.100
marble, over the total
number of outcomes.

00:01:13.100 --> 00:01:15.720
So how many ways are there-- how
many events-- you're going

00:01:15.720 --> 00:01:16.710
to pick out a ball.

00:01:16.710 --> 00:01:19.980
How many of those-- how many
ways-- or how many balls can

00:01:19.980 --> 00:01:21.670
you pick out that
will be yellow?

00:01:21.670 --> 00:01:23.710
Well, there's three balls.

00:01:23.710 --> 00:01:25.580
There's three balls that you
can pick out as yellow.

00:01:25.580 --> 00:01:26.400
One, two, and three.

00:01:26.400 --> 00:01:28.260
We're assuming that they're
all equally likely

00:01:28.260 --> 00:01:28.970
to be picked up.

00:01:28.970 --> 00:01:30.090
They're all mixed in there.

00:01:30.090 --> 00:01:32.760
So there's three yellow balls
you could pick up.

00:01:32.760 --> 00:01:35.080
Now how many total
balls are there?

00:01:35.080 --> 00:01:37.810
How many total balls are
there to be picked up?

00:01:37.810 --> 00:01:40.540
Well, you could be picking up
any one of eight balls, right?

00:01:40.540 --> 00:01:43.010
One, two, three, four, five,
six, seven, eight.

00:01:43.010 --> 00:01:45.060
So there's a total number
of eight balls.

00:01:45.060 --> 00:01:49.160
So the probability of picking
a yellow marble is 3/8.

00:01:49.160 --> 00:01:52.730
So you have a 3/8 chance of
picking a yellow marble.

00:01:52.730 --> 00:01:53.500
And just a way to
think about it.

00:01:53.500 --> 00:01:54.560
I just want to reiterate.

00:01:54.560 --> 00:01:57.840
It's just the number
of outcomes.

00:01:57.840 --> 00:01:59.490
There's eight total outcomes.

00:01:59.490 --> 00:02:01.320
There's eight different
things that you could

00:02:01.320 --> 00:02:02.580
pull out of the bag.

00:02:02.580 --> 00:02:04.840
Three of them satisfy
your conditions.

00:02:04.840 --> 00:02:10.080
Three of them would be
a yellow marble.


Deductive reasoning 2

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.390
So they tell us that Hiram
solved the equation, 5 plus

00:00:04.390 --> 00:00:08.930
the square root of x plus 14
is equal to x plus 7, using

00:00:08.930 --> 00:00:09.800
the following steps.

00:00:09.800 --> 00:00:10.950
Let's see what he did.

00:00:10.950 --> 00:00:12.830
He subtracted 5 from
both sides.

00:00:12.830 --> 00:00:13.860
Yeah, that's reasonable
enough.

00:00:13.860 --> 00:00:15.030
That got that 5 out
of the way.

00:00:15.030 --> 00:00:17.120
And then 7 minus 5 is 2.

00:00:17.120 --> 00:00:18.850
And then he squared
both sides.

00:00:18.850 --> 00:00:22.690
The square root of x plus 14
squared is just x plus 14.

00:00:22.690 --> 00:00:23.310
That makes sense.

00:00:23.310 --> 00:00:26.700
And x plus 2 squared is
x plus 2 squared.

00:00:26.700 --> 00:00:31.040
And then he uses the pattern for
square binomials to expand

00:00:31.040 --> 00:00:31.690
the right-hand side.

00:00:31.690 --> 00:00:35.220
OK, so he just multiplied out
x plus 2, times x plus 2, to

00:00:35.220 --> 00:00:37.650
get x squared plus 4x, plus 4.

00:00:37.650 --> 00:00:40.975
Then he subtracted x plus 14
from both sides, so he gets a

00:00:40.975 --> 00:00:41.830
0 on the left-hand side.

00:00:41.830 --> 00:00:44.260
So when you take x from
4x, you get 3x.

00:00:44.260 --> 00:00:48.650
When you take 14 from 4
you get negative 10.

00:00:48.650 --> 00:00:49.840
So that all makes sense.

00:00:49.840 --> 00:00:51.450
Then he factored the
right-hand side.

00:00:51.450 --> 00:00:54.580
Let's see, 5 times negative
2 is negative 10.

00:00:54.580 --> 00:00:56.540
5 plus negative 2 is 3.

00:00:56.540 --> 00:00:57.320
That makes sense.

00:00:57.320 --> 00:00:59.480
Then he uses a 0 product
property

00:00:59.480 --> 00:01:00.280
to solve the equation.

00:01:00.280 --> 00:01:02.330
That's just this property--
look, if you have two things

00:01:02.330 --> 00:01:05.950
and you take their product and
it equals 0, one or both of

00:01:05.950 --> 00:01:07.610
them must be equal to 0.

00:01:07.610 --> 00:01:10.920
So that means x plus 5 is equal
to 0, or x minus 2 is

00:01:10.920 --> 00:01:11.950
equal to 0.

00:01:11.950 --> 00:01:14.660
And so if x plus 5 is equal
to 0, that's x is equal to

00:01:14.660 --> 00:01:15.710
negative 5.

00:01:15.710 --> 00:01:19.190
x is equal to 2 of 0, that's
x is equal to 2.

00:01:19.190 --> 00:01:20.550
Let me just write
that down here.

00:01:20.550 --> 00:01:23.070
So all he did in this step
is he says, x plus

00:01:23.070 --> 00:01:24.700
5 is equal to 0.

00:01:24.700 --> 00:01:27.210
Or x minus 2 is equal to 0.

00:01:27.210 --> 00:01:31.320
From this you get this right
over here, because you

00:01:31.320 --> 00:01:32.670
subtract 5 from both sides.

00:01:32.670 --> 00:01:35.620
And then from this you get
that right over there.

00:01:35.620 --> 00:01:36.820
Then he checked both answers.

00:01:36.820 --> 00:01:39.200
He substituted negative 5 into
the original equation.

00:01:39.200 --> 00:01:41.762
So he substituted negative
5 in there.

00:01:41.762 --> 00:01:43.230
It shows up twice.

00:01:43.230 --> 00:01:45.610
5 plus the square root of
negative 5, plus 14 is equal

00:01:45.610 --> 00:01:47.370
to negative 5 plus 7.

00:01:47.370 --> 00:01:48.410
Let's see what else he did.

00:01:48.410 --> 00:01:50.405
Then this becomes square
root of 9.

00:01:50.405 --> 00:01:51.420
This becomes 2.

00:01:51.420 --> 00:01:54.070
Then you get 5 plus 3 is equal
to 2, which is false.

00:01:54.070 --> 00:01:55.280
This is not true.

00:01:55.280 --> 00:01:56.790
And he wrote that down.

00:01:56.790 --> 00:01:58.430
And then he tried out 2.

00:01:58.430 --> 00:01:59.740
His other solution.

00:01:59.740 --> 00:02:02.950
When you substitute 2 you get
2 plus 14, which is 16.

00:02:02.950 --> 00:02:03.960
2 plus 7 is 9.

00:02:03.960 --> 00:02:05.340
Square root of 16 is 4.

00:02:05.340 --> 00:02:07.410
And this is the principal root
of 16 we're talking about, so

00:02:07.410 --> 00:02:08.940
we're taking the positive
square root.

00:02:08.940 --> 00:02:11.150
And then 5 plus 4
is equal to 9.

00:02:11.150 --> 00:02:12.900
So this works out.

00:02:12.900 --> 00:02:14.785
And then he says that
the answer is x

00:02:14.785 --> 00:02:16.590
equals 2, which is right.

00:02:16.590 --> 00:02:19.810
Now this whole exercise, all
they want us to know is, is

00:02:19.810 --> 00:02:22.900
this an example of deductive
reasoning?

00:02:22.900 --> 00:02:24.210
Explain.

00:02:24.210 --> 00:02:27.760
And it is an example of
deductive reasoning.

00:02:27.760 --> 00:02:32.540
He started off with a known
statement, with a known-- we

00:02:32.540 --> 00:02:35.160
could call that a known
fact-- if we assume

00:02:35.160 --> 00:02:36.070
that that's a fact.

00:02:36.070 --> 00:02:37.460
He started off with that.

00:02:37.460 --> 00:02:40.440
And just doing logical
operations, he was able to

00:02:40.440 --> 00:02:42.370
deduce, step by step,
he was able to

00:02:42.370 --> 00:02:43.860
manipulate other truths.

00:02:43.860 --> 00:02:47.020
He started with a fact and using
logical operations he

00:02:47.020 --> 00:02:49.050
was able to come up
with other facts.

00:02:49.050 --> 00:02:51.580
And go all the way down here
and then check his answers,

00:02:51.580 --> 00:02:54.680
and eventually come up with
the notion that if this is

00:02:54.680 --> 00:02:57.930
true, then this must
also be true.

00:02:57.930 --> 00:02:59.420
So that is deductive
reasoning.

00:02:59.420 --> 00:03:03.020
You start with facts, use
logical steps or operations,

00:03:03.020 --> 00:03:06.030
or logical reasoning to come
up with other facts.

00:03:06.030 --> 00:03:07.200
He's not estimating.

00:03:07.200 --> 00:03:08.200
He's not generalizing.

00:03:08.200 --> 00:03:10.350
He's not assuming some
trend will continue.

00:03:10.350 --> 00:03:12.310
He started with something he
knows is true and gets to

00:03:12.310 --> 00:03:13.990
something else he
knows is true.

00:03:13.990 --> 00:03:15.010
And this is a bit of a review.

00:03:15.010 --> 00:03:17.520
You're probably wondering
why the

00:03:17.520 --> 00:03:19.480
negative 5 wasn't a solution.

00:03:19.480 --> 00:03:21.780
And as a little bit of a hint
here-- and I'll let you think

00:03:21.780 --> 00:03:25.050
about why it didn't end up as a
solution-- when we took the

00:03:25.050 --> 00:03:27.460
square root of 9 here, we took
the positive square root.

00:03:27.460 --> 00:03:30.110
And when anyone just says a
square root like that, that

00:03:30.110 --> 00:03:31.710
means the positive
square root, the

00:03:31.710 --> 00:03:33.330
principal square root.

00:03:33.330 --> 00:03:35.670
But if we were to take the
negative square root here,

00:03:35.670 --> 00:03:37.640
this equation would
have held up.

00:03:37.640 --> 00:03:41.200
Because 5 plus negative
3 is equal to 2.

00:03:41.200 --> 00:03:43.780
So I'll let you think about at
what step of this equation

00:03:43.780 --> 00:03:46.530
would a negative number
have worked?

00:03:46.530 --> 00:03:47.810
And it has something
to do with when

00:03:47.810 --> 00:03:49.200
we square both sides.

00:03:49.200 --> 00:03:50.890
But that has nothing to do with
the actual question at

00:03:50.890 --> 00:03:52.190
hand, so I'll leave you there.

00:03:52.190 --> 00:03:54.360
We explained it before,
in the past. So this

00:03:54.360 --> 00:03:56.480
was deductive reasoning.


Difference between inductive and deductive reasoning

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.740
Think about this real
world problem.

00:00:02.740 --> 00:00:06.660
To estimate the population of a
town in upcoming years, one

00:00:06.660 --> 00:00:09.670
of the town workers collected
populations from past years

00:00:09.670 --> 00:00:11.000
and made this table.

00:00:11.000 --> 00:00:14.210
So they have years starting
with 1950 and going up by

00:00:14.210 --> 00:00:15.310
increments of 10 years.

00:00:15.310 --> 00:00:18.130
And they just figure out the
population of the town in each

00:00:18.130 --> 00:00:19.400
of those years.

00:00:19.400 --> 00:00:22.040
The town wants to estimate
the population for

00:00:22.040 --> 00:00:25.950
2015, 2018, and 2020.

00:00:25.950 --> 00:00:29.930
To do this, will you be using
inductive reasoning or

00:00:29.930 --> 00:00:31.510
deductive reasoning?

00:00:31.510 --> 00:00:33.100
Now before answering that, let's
just think about what

00:00:33.100 --> 00:00:35.850
inductive reasoning is and what
deductive reasoning is.

00:00:35.850 --> 00:00:38.530
Inductive reasoning is looking
for a pattern or

00:00:38.530 --> 00:00:39.940
looking for a trend.

00:00:39.940 --> 00:00:43.220
So it's looking for a trend
or a pattern and then

00:00:43.220 --> 00:00:45.510
generalizing.

00:00:45.510 --> 00:00:47.640
You could imagine, it's kind
of extrapolating the

00:00:47.640 --> 00:00:51.350
information you have,
generalizing.

00:00:51.350 --> 00:00:55.470
Deductive reasoning is taking
some set of data or some set

00:00:55.470 --> 00:00:59.170
of facts and using that to
come up with other, or

00:00:59.170 --> 00:01:01.790
deducing some other, facts
that you know are true.

00:01:01.790 --> 00:01:04.230
When you generalize you don't
know necessarily whether the

00:01:04.230 --> 00:01:06.740
trend will continue, but
you assume it will.

00:01:06.740 --> 00:01:08.850
You don't know 100%
it'll be true.

00:01:08.850 --> 00:01:11.050
With deductive reasoning,
you know it'll be true.

00:01:11.050 --> 00:01:14.300
You're starting with facts.

00:01:14.300 --> 00:01:17.250
And then you're deducing other
facts from those facts.

00:01:17.250 --> 00:01:19.200
Other facts that you
know are true.

00:01:19.200 --> 00:01:22.230
Now, when the town is
estimating, and that's kind of

00:01:22.230 --> 00:01:24.570
a clue that they're not going
to know the exact population

00:01:24.570 --> 00:01:26.990
for 2015, 2018, and 2020.

00:01:26.990 --> 00:01:29.060
But when they estimate the
population for these years,

00:01:29.060 --> 00:01:30.670
these years are in the future.

00:01:30.670 --> 00:01:33.120
They don't know what
the population

00:01:33.120 --> 00:01:34.370
will be in those years.

00:01:34.370 --> 00:01:36.900
The best they can do is see
the trend of growth here,

00:01:36.900 --> 00:01:40.050
maybe see what percentage it's
growing every 10 years.

00:01:40.050 --> 00:01:42.720
Or what the absolute number of
growth is every 10 years.

00:01:42.720 --> 00:01:44.730
And maybe they can continue
that trend out to

00:01:44.730 --> 00:01:47.740
2015, 2018, and 2020.

00:01:47.740 --> 00:01:50.320
So they'll be looking for trend
in growth over this

00:01:50.320 --> 00:01:51.480
60-year period.

00:01:51.480 --> 00:01:53.360
And then they'll generalize
it out to these years.

00:01:53.360 --> 00:01:55.470
So this will be inductive
reasoning.


Inductive reasoning 3

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.030
Luis looked at the expression x
minus y times x squared plus

00:00:05.030 --> 00:00:08.410
xy plus y squared, and
wrote the following.

00:00:08.410 --> 00:00:10.600
He said, using the distributive
property-- so he

00:00:10.600 --> 00:00:14.360
took x minus y and he
distributed this expression

00:00:14.360 --> 00:00:16.690
onto the x and this expression
onto the negative y.

00:00:16.690 --> 00:00:19.010
So that's why we have an
x times this entire

00:00:19.010 --> 00:00:20.585
expression over here.

00:00:20.585 --> 00:00:24.550
x times the entire expression
minus y times this entire

00:00:24.550 --> 00:00:25.610
expression.

00:00:25.610 --> 00:00:27.800
Then he says, using the
distributive property again,

00:00:27.800 --> 00:00:30.290
this is equal to-- so then he
distributed this x into that

00:00:30.290 --> 00:00:36.010
and got all of the first
three terms here.

00:00:36.010 --> 00:00:38.330
He got these first three
terms. And then he

00:00:38.330 --> 00:00:39.990
distributed the y.

00:00:39.990 --> 00:00:43.940
He distributed the y over here
and he got these three terms.

00:00:43.940 --> 00:00:47.760
And then he saw, looks like
that this term x squared y

00:00:47.760 --> 00:00:50.710
cancels out with a minus x
squared y, and the xy squared

00:00:50.710 --> 00:00:53.770
cancels out with the negative
xy squared.

00:00:53.770 --> 00:00:55.480
Then he saw that it equals
x to the third

00:00:55.480 --> 00:00:57.250
minus y to the third.

00:00:57.250 --> 00:01:00.290
And he wrote down, I conjecture
that x minus y

00:01:00.290 --> 00:01:03.990
times x squared plus xy plus y
squared is equal to x to the

00:01:03.990 --> 00:01:07.220
third minus y to the third
for all x and y.

00:01:07.220 --> 00:01:09.960
Did Carlos use inductive
reasoning?

00:01:09.960 --> 00:01:11.230
Explain.

00:01:11.230 --> 00:01:14.410
Well, inductive reasoning is
looking at a sample of

00:01:14.410 --> 00:01:19.620
things-- looking at some set
of data, if you will-- and

00:01:19.620 --> 00:01:22.340
then making a generalization
based on that.

00:01:22.340 --> 00:01:24.850
You're not 100% sure, but based
on what you've seen so

00:01:24.850 --> 00:01:27.280
far, you think that the pattern
would continue.

00:01:27.280 --> 00:01:30.720
Or you think it might be true
for all things that have that

00:01:30.720 --> 00:01:32.240
type of property or whatever.

00:01:32.240 --> 00:01:34.050
Now in this situation,
he didn't look at

00:01:34.050 --> 00:01:35.220
some type of a sample.

00:01:35.220 --> 00:01:36.630
He actually just did a proof.

00:01:36.630 --> 00:01:38.460
He multiplied this out
algebraically.

00:01:38.460 --> 00:01:41.630
In fact, it's incorrect for him
to say I conjecture here.

00:01:41.630 --> 00:01:46.650
A conjecture is a statement or
proposition that is unproven,

00:01:46.650 --> 00:01:48.470
but it's probably going
to be true.

00:01:48.470 --> 00:01:51.020
It's unproven but it seems
reasonable, or it seems likely

00:01:51.020 --> 00:01:51.900
that it's true.

00:01:51.900 --> 00:01:53.030
This isn't a conjecture.

00:01:53.030 --> 00:01:53.950
This is proven.

00:01:53.950 --> 00:01:58.930
He proved that x minus y times
x squared plus xy plus y

00:01:58.930 --> 00:02:02.100
squared is equal to x to the
third minus y third.

00:02:02.100 --> 00:02:04.240
He should have written-- and
this is a much stronger thing

00:02:04.240 --> 00:02:10.320
to say-- he should have said, I
proved that this is true for

00:02:10.320 --> 00:02:11.340
all x and y.

00:02:11.340 --> 00:02:14.010
So to answer the question, did
he use inductive reasoning?

00:02:14.010 --> 00:02:14.350
No.

00:02:14.350 --> 00:02:17.000
I would say that he made
an outright proof.

00:02:17.000 --> 00:02:22.730
No, he made a proof.

00:02:22.730 --> 00:02:26.010
Inductive reasoning would have
been, if he would have saw, if

00:02:26.010 --> 00:02:30.290
you would have given him 5 minus
2 is equal to-- or 5

00:02:30.290 --> 00:02:35.150
minus 2 times 5 squared plus 5
times 2 plus 2 squared, and

00:02:35.150 --> 00:02:37.180
you saw that that was equal to
the same thing as 5 to the

00:02:37.180 --> 00:02:38.630
third minus 2 to the third.

00:02:38.630 --> 00:02:40.770
And then let's say he did it
for, I don't know, one in

00:02:40.770 --> 00:02:42.350
seven in a couple of examples.

00:02:42.350 --> 00:02:44.430
And it kept holding for all the
examples that it was the

00:02:44.430 --> 00:02:46.900
first number cubed minus the
second number cubed.

00:02:46.900 --> 00:02:48.340
Then it would have been
inductive reasoning to say

00:02:48.340 --> 00:02:51.140
that that is true for
all numbers x and y.

00:02:51.140 --> 00:02:52.690
But here it's not
an induction.

00:02:52.690 --> 00:02:53.890
He didn't use induction.

00:02:53.890 --> 00:02:54.930
Or I shouldn't say induction.

00:02:54.930 --> 00:02:56.560
He didn't use inductive
reasoning.

00:02:56.560 --> 00:03:00.500
He outright proved that
this statement is true

00:03:00.500 --> 00:03:02.660
for all x and y.


Inductive reasoning 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.050
Jill looked at the following
sequence.

00:00:03.050 --> 00:00:07.810
0, 3, 8, 15, 24, 35.

00:00:07.810 --> 00:00:10.190
And it just keeps going, I
guess, with a dot, dot, dot.

00:00:10.190 --> 00:00:12.410
She saw that the numbers
were each 1 less

00:00:12.410 --> 00:00:14.110
than a square number.

00:00:14.110 --> 00:00:16.820
0 is 1 less than 1, which
is a square number.

00:00:16.820 --> 00:00:18.160
3 is 1 less than 4.

00:00:18.160 --> 00:00:19.740
8 is 1 less than 9.

00:00:19.740 --> 00:00:20.950
15 is 1 less than 16.

00:00:20.950 --> 00:00:22.960
Yeah, they were all 1 less
than a square number.

00:00:22.960 --> 00:00:26.030
And conjectured that the
nth number would be n

00:00:26.030 --> 00:00:27.180
squared minus 1.

00:00:27.180 --> 00:00:29.850
Now conjecture, that sounds
like a very fancy word.

00:00:29.850 --> 00:00:32.600
When someone makes a conjecture,
they conjecture,

00:00:32.600 --> 00:00:35.260
that just means that they're
making a statement that seems,

00:00:35.260 --> 00:00:37.470
or they're making a proposition
that seems likely

00:00:37.470 --> 00:00:38.130
to be true.

00:00:38.130 --> 00:00:40.060
It seems like a very reasonable
thing to say.

00:00:40.060 --> 00:00:41.550
But it's not definitely true.

00:00:41.550 --> 00:00:44.250
So she conjectured that the
nth number would be n

00:00:44.250 --> 00:00:47.300
squared minus 1.

00:00:47.300 --> 00:00:51.010
The reason why this is a
conjecture as opposed to a

00:00:51.010 --> 00:00:53.960
100% definitely true statement,
is we don't know

00:00:53.960 --> 00:00:55.870
whether this pattern
continues.

00:00:55.870 --> 00:00:58.980
She's just going off of the
pattern that she saw so far

00:00:58.980 --> 00:01:00.100
and she just generalized it.

00:01:00.100 --> 00:01:02.690
She just assumes that
it keeps on going.

00:01:02.690 --> 00:01:05.099
But we don't know whether it
necessarily keeps going.

00:01:05.099 --> 00:01:09.620
Maybe the next number, you would
expect it to be 48, but

00:01:09.620 --> 00:01:10.690
maybe it's not 48.

00:01:10.690 --> 00:01:11.880
Maybe it's something weird.

00:01:11.880 --> 00:01:12.780
Maybe it's 2.

00:01:12.780 --> 00:01:14.270
Maybe it's 500.

00:01:14.270 --> 00:01:16.870
And so the conjecture wouldn't
hold up if you were to see

00:01:16.870 --> 00:01:20.020
that, but based on the evidence
you see so far it

00:01:20.020 --> 00:01:21.600
seems completely reasonable
that this

00:01:21.600 --> 00:01:23.030
pattern would continue.

00:01:23.030 --> 00:01:25.690
And so she conjectured that
the nth number would be n

00:01:25.690 --> 00:01:27.230
squared minus 1.

00:01:27.230 --> 00:01:28.510
Completely reasonable.

00:01:28.510 --> 00:01:31.195
Now did Jill use inductive
reasoning?

00:01:34.070 --> 00:01:35.860
Yes, she used inductive
reasoning.

00:01:35.860 --> 00:01:38.200
That's what inductive
reasoning is.

00:01:38.200 --> 00:01:39.640
You see a pattern.

00:01:39.640 --> 00:01:46.160
In this case, every term in this
sequence so far was-- if

00:01:46.160 --> 00:01:48.990
it's the third term, it
was 3 squared minus 1.

00:01:48.990 --> 00:01:50.810
The fourth term is 4
squared minus 1.

00:01:50.810 --> 00:01:54.370
The fifth term is 5
squared minus 1.

00:01:54.370 --> 00:01:56.980
So she saw the pattern and she
just generalized it to say,

00:01:56.980 --> 00:02:00.730
well, I think or I've
conjectured that the nth

00:02:00.730 --> 00:02:02.770
number will be n squared
minus 1.

00:02:02.770 --> 00:02:05.160
That's what inductive reasoning
is all about.

00:02:05.160 --> 00:02:08.160
You're not always going to be
100%, or you definitely won't

00:02:08.160 --> 00:02:10.660
be 100% sure that you're right,
that the nth number

00:02:10.660 --> 00:02:12.160
will be n squared minus 1.

00:02:12.160 --> 00:02:14.200
But based on the pattern you've
seen so far, it's a

00:02:14.200 --> 00:02:17.160
completely reasonable thing
to-- I guess you

00:02:17.160 --> 00:02:18.410
could say-- to induce.