Representing data with matrices

WEBVTT
Kind: captions
Language: en

00:00:00.501 --> 00:00:02.678
Voiceover:The price of
things at two supermarkets

00:00:02.678 --> 00:00:04.388
are different in different cities.

00:00:04.388 --> 00:00:08.562
Toilet paper in Duluth,
Minnesota cost 3.99 a package

00:00:08.562 --> 00:00:13.142
while toilet paper in New
York City cost 8.95 a package.

00:00:13.142 --> 00:00:17.281
In Duluth, toothpaste costs $1.95 a tube

00:00:17.281 --> 00:00:22.086
while in New York City
it costs $5.25 a tube.

00:00:22.086 --> 00:00:23.993
The data for this can be encoded

00:00:23.993 --> 00:00:26.316
in the following grocery matrix.

00:00:26.316 --> 00:00:27.972
Let's see if this makes sense.

00:00:29.095 --> 00:00:33.241
They have the price of
toilet paper in Duluth 3.99

00:00:33.241 --> 00:00:35.356
that's this right over here.

00:00:35.356 --> 00:00:37.938
Then they have the price of toilet paper

00:00:37.938 --> 00:00:40.079
in New York at 8.95

00:00:40.079 --> 00:00:41.896
so that's this right over here.

00:00:41.896 --> 00:00:44.093
This first row right over here,

00:00:44.093 --> 00:00:45.587
this is the toilet paper row.

00:00:45.587 --> 00:00:50.510
I'll just write TP for short,
that's the toilet paper row

00:00:50.510 --> 00:00:53.910
and this first column
looks like it's Duluth

00:00:53.910 --> 00:00:58.725
while this second column
looks like it's New York City.

00:01:00.354 --> 00:01:02.624
Let's see if that works
out for the other data.

00:01:02.624 --> 00:01:08.654
In Duluth a toothpaste cost $1.95 a tube,

00:01:08.654 --> 00:01:12.778
so this data right over
here, is right over here.

00:01:12.778 --> 00:01:14.931
This looks like the toothpaste row.

00:01:14.931 --> 00:01:18.683
Toothpaste actually is also TP

00:01:18.683 --> 00:01:20.132
so I'll just write out tooth.

00:01:21.870 --> 00:01:23.098
This is the toothpaste row,

00:01:23.098 --> 00:01:24.981
this is toothpaste in Duluth, $1.95.

00:01:24.981 --> 00:01:28.102
Toothpaste in New York City, 5.25.

00:01:28.102 --> 00:01:31.080
That's how they have set-up
the data right over here.

00:01:31.080 --> 00:01:36.785
Which statements are true
about the above matrix?

00:01:36.785 --> 00:01:38.359
Select all that apply.

00:01:38.359 --> 00:01:41.554
The following matrix can also be used

00:01:41.554 --> 00:01:43.560
to contain the same information as G.

00:01:43.560 --> 00:01:45.459
That's what's interesting
about the matrix,

00:01:45.459 --> 00:01:46.928
what we have right over here

00:01:46.928 --> 00:01:49.584
is essentially an encapsulation
of all of the data

00:01:49.584 --> 00:01:51.911
that we have in this upper paragraph

00:01:51.911 --> 00:01:53.857
and it's useful because a computer

00:01:53.857 --> 00:01:55.593
could make use of this data

00:01:55.593 --> 00:01:57.626
as long as it knows what these rows

00:01:57.626 --> 00:01:59.705
and what these columns actually represent.

00:01:59.705 --> 00:02:01.657
The first thing they ask us

00:02:01.657 --> 00:02:04.674
is whether the following
matrix can also be used

00:02:04.674 --> 00:02:06.553
to contain the same information.

00:02:06.553 --> 00:02:10.899
Let's see, this is the price
of toilet paper in Duluth

00:02:10.899 --> 00:02:13.931
and this right over here is the price

00:02:13.931 --> 00:02:17.074
of toilet paper in New York City.

00:02:18.043 --> 00:02:19.709
They have the exact same first row,

00:02:19.709 --> 00:02:22.391
I would say that looks
like the toilet paper row

00:02:22.391 --> 00:02:24.658
and that would be Duluth column

00:02:24.658 --> 00:02:26.837
and this would be the
New York City column.

00:02:26.837 --> 00:02:30.105
Then if we define things that way

00:02:30.105 --> 00:02:32.885
then this would have to
be the toothpaste row,

00:02:35.315 --> 00:02:37.757
but the price of toothpaste in Duluth

00:02:37.757 --> 00:02:40.017
is not 5.25,

00:02:40.017 --> 00:02:42.609
and the price of
toothpaste in New York City

00:02:42.609 --> 00:02:44.120
is not 1.95.

00:02:44.120 --> 00:02:47.525
This one is not representing
the same information

00:02:47.525 --> 00:02:48.555
as that up there.

00:02:50.678 --> 00:02:52.677
You can't just randomly order this around.

00:02:52.677 --> 00:02:54.432
Now you could represent it in other ways,

00:02:54.432 --> 00:02:56.007
you could have another ...

00:02:56.007 --> 00:02:58.057
Let's say I have the matrix A,

00:02:58.057 --> 00:03:01.092
we could have picked to
do something like this

00:03:01.092 --> 00:03:03.016
where we could have said

00:03:03.016 --> 00:03:07.075
well maybe this is the toothpaste column

00:03:08.721 --> 00:03:11.104
and maybe this is the toilet paper column,

00:03:11.104 --> 00:03:13.808
and that this first row is New York City

00:03:13.808 --> 00:03:16.544
and the second row is Duluth.

00:03:16.544 --> 00:03:18.254
You could have done something like this

00:03:18.254 --> 00:03:21.321
toothpaste in New York City was 5.25,

00:03:21.321 --> 00:03:25.435
toilet paper in New York City is 8.95,

00:03:27.111 --> 00:03:29.713
toothpaste in Duluth is 1.95

00:03:29.713 --> 00:03:34.868
and toilet paper in Duluth is 3.99.

00:03:34.868 --> 00:03:38.251
This would have been, this matrix A

00:03:38.251 --> 00:03:39.459
that I've just constructed,

00:03:39.459 --> 00:03:41.906
this does contain the same data

00:03:41.906 --> 00:03:45.225
because if we appropriately
define our columns and rows

00:03:45.225 --> 00:03:47.043
to represent toothpaste or toilet paper

00:03:47.043 --> 00:03:48.204
in New York City or Duluth,

00:03:48.204 --> 00:03:49.852
you can contain that information.

00:03:49.852 --> 00:03:52.479
The problem with this one
is it's not consistent.

00:03:52.479 --> 00:03:54.124
This first row makes us think,

00:03:54.124 --> 00:03:56.518
okay this must be the Duluth column,

00:03:56.518 --> 00:03:58.282
this must be the New York City column

00:03:58.282 --> 00:04:00.990
and that the second row must be toothpaste

00:04:00.990 --> 00:04:02.571
since the first row is toilet paper

00:04:02.571 --> 00:04:04.171
but then they got things mixed up.

00:04:04.171 --> 00:04:06.188
If they switch these two things around

00:04:06.188 --> 00:04:07.846
then obviously you get the same matrix

00:04:07.846 --> 00:04:10.174
but then at least the data
would have been consistent.

00:04:10.174 --> 00:04:12.498
I definitely would not
say that this matrix

00:04:12.498 --> 00:04:14.504
contains the same information.

00:04:14.504 --> 00:04:17.778
The second column represents
the price of toothpaste

00:04:17.778 --> 00:04:19.425
in the two cities.

00:04:19.425 --> 00:04:22.674
The second column is this right over here,

00:04:22.674 --> 00:04:24.754
no, that's not the price of
toothpaste in the two cities,

00:04:24.754 --> 00:04:26.823
that's the price of
the two different goods

00:04:26.823 --> 00:04:27.804
in New York City,

00:04:27.804 --> 00:04:29.675
so I wouldn't check that one either.

00:04:29.675 --> 00:04:34.875
Element G two comma one is equal to 5.25,

00:04:34.875 --> 00:04:36.890
so when someone says ...

00:04:36.890 --> 00:04:38.547
If you have the matrix

00:04:38.547 --> 00:04:41.110
and they give you two comma one.

00:04:41.110 --> 00:04:44.573
These are the indices of
the row and the column,

00:04:44.573 --> 00:04:47.920
so this is a second row, first column.

00:04:47.920 --> 00:04:52.151
Let's go there, second
row so this is one, two,

00:04:52.151 --> 00:04:57.152
second row, first column
is right over there.

00:04:57.152 --> 00:05:08.094
G two comma one is not 5.25,

00:05:08.094 --> 00:05:10.037
so I wouldn't check that one either.

00:05:10.037 --> 00:05:12.844
A change in the price of
toilet paper in Duluth

00:05:12.844 --> 00:05:14.366
can be represented by the matrix.

00:05:14.366 --> 00:05:19.228
Toilet paper in Duluth is this entry

00:05:19.228 --> 00:05:20.990
of our matrix right over there.

00:05:20.990 --> 00:05:23.526
Change in toilet paper in Duluth

00:05:23.526 --> 00:05:26.496
would represent a change in this 3.99

00:05:26.496 --> 00:05:28.414
but they didn't change the 3.99,

00:05:28.414 --> 00:05:30.374
they changed the 9.75.

00:05:30.374 --> 00:05:32.509
The 9.75 would represent a change

00:05:32.509 --> 00:05:35.047
in the price of toilet
paper in New York City.

00:05:35.047 --> 00:05:36.820
I wouldn't check that one either,

00:05:36.820 --> 00:05:42.147
so I would only go with none of the above.


Comparing insurance with expected value

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:03.966
Voiceover:Paul has the option
between a high-deductible

00:00:03.966 --> 00:00:09.234
or a low-deductible health insurance plan.

00:00:09.234 --> 00:00:11.433
When we talk about the
deductible in health insurance,

00:00:11.433 --> 00:00:12.700
if someone says that they have a plan

00:00:12.700 --> 00:00:14.767
with $1,000 deductible,

00:00:14.767 --> 00:00:16.500
that means that the insurance company

00:00:16.500 --> 00:00:19.967
only pays the medical cost
after the first $1,000.

00:00:19.967 --> 00:00:21.766
So if you have a $1,000 deductible

00:00:21.766 --> 00:00:24.900
and you, say, incur medical cost of $800,

00:00:24.900 --> 00:00:26.767
you're going to pay that $800.

00:00:26.767 --> 00:00:28.500
The insurance company won't pay anything.

00:00:28.500 --> 00:00:30.233
If you have a deductible of $1,000

00:00:30.233 --> 00:00:33.367
and your total medical costs are $1,200,

00:00:33.367 --> 00:00:35.300
you're going to pay the first $1,000,

00:00:35.300 --> 00:00:38.100
and then the insurance company
will kick in after that.

00:00:38.100 --> 00:00:39.100
So with that out of the way,

00:00:39.100 --> 00:00:40.566
let's think about his 2 plans.

00:00:40.566 --> 00:00:42.771
If Paul chooses the low-deductible plan,

00:00:42.771 --> 00:00:45.646
he will have to pay the first $1,000

00:00:45.646 --> 00:00:46.367
of any ...

00:00:46.367 --> 00:00:47.434
let me do that in purple,

00:00:47.434 --> 00:00:50.766
the first $1,000 of any medical cost.

00:00:50.766 --> 00:00:54.433
The low-deductible plan
costs $8,000 for a year.

00:00:54.433 --> 00:00:56.766
So in this situation,
he's going to pay $8,000

00:00:56.766 --> 00:00:58.433
to get the insurance.

00:00:58.433 --> 00:01:01.167
If he has $900 of medical expenses,

00:01:01.167 --> 00:01:03.100
the insurance company still pays nothing.

00:01:03.100 --> 00:01:06.235
If he has $2,000 of medical expenses,

00:01:06.235 --> 00:01:08.300
then he pays the first $1,000,

00:01:08.300 --> 00:01:09.433
and then the insurance company would pay

00:01:09.433 --> 00:01:10.566
the next $1,000.

00:01:10.566 --> 00:01:12.900
If he has $10,000 in medical expenses,

00:01:12.900 --> 00:01:14.300
he would pay the first $1,000,

00:01:14.300 --> 00:01:15.433
and then the insurance company would pay

00:01:15.433 --> 00:01:17.401
the next $9,000.

00:01:17.901 --> 00:01:20.433
If Paul chooses the high-deductible plan,

00:01:20.433 --> 00:01:23.433
he will have to pay the first $2,500

00:01:23.433 --> 00:01:24.966
of any medical cost.

00:01:24.966 --> 00:01:28.033
The high-deductible plan
costs $7,500 a year.

00:01:28.033 --> 00:01:30.233
It makes sense that the
high-deductible plan

00:01:30.233 --> 00:01:32.108
costs less than the low-deductible plan

00:01:32.108 --> 00:01:34.567
because here, the
insurance doesn't kick in

00:01:34.567 --> 00:01:39.103
until he has over $2,500
of medical expenses,

00:01:39.103 --> 00:01:40.900
while here, it was only $1,000.

00:01:40.900 --> 00:01:43.100
To help himself choose a plan,

00:01:43.100 --> 00:01:45.966
Paul found some statistics
about common health problems

00:01:45.966 --> 00:01:47.834
for people similar to him.

00:01:47.834 --> 00:01:49.367
Assume that the table below,

00:01:49.367 --> 00:01:50.766
and I put it up here on the right,

00:01:50.766 --> 00:01:53.500
correctly shows the probabilities in cost

00:01:53.500 --> 00:01:58.035
of total medical incidents
within the next year.

00:01:58.035 --> 00:01:59.437
So this right over here, it's saying

00:01:59.437 --> 00:02:01.700
what's the probability he
has $0 in medical costs?

00:02:01.700 --> 00:02:02.768
What's the probability of ...

00:02:02.768 --> 00:02:05.700
he has a 25% probability of $1,000,

00:02:05.700 --> 00:02:07.740
20% probability, $4,000.

00:02:07.940 --> 00:02:10.103
And this is a simplification,

00:02:10.103 --> 00:02:12.966
a pretty dramatic simplification
from the real world.

00:02:12.966 --> 00:02:15.300
In the real world, the
way this makes it sound

00:02:15.300 --> 00:02:17.900
is there's only 5 possible medical costs

00:02:17.900 --> 00:02:19.167
that someone might have,

00:02:19.167 --> 00:02:22.502
$0, $1,000, $4,000, $7,000, and $15,000.

00:02:22.502 --> 00:02:26.033
In the real world, you could
have $20 of medical cost.

00:02:26.033 --> 00:02:27.300
You could have $20,000.

00:02:27.300 --> 00:02:29.835
You could have $999.

00:02:29.835 --> 00:02:32.300
So in the real world,
there is many, many more

00:02:32.300 --> 00:02:35.300
situations here that you
would have to redistribute

00:02:35.300 --> 00:02:36.901
the probabilities accordingly.

00:02:36.901 --> 00:02:39.434
But with that said, this
isn't a bad approximation.

00:02:39.434 --> 00:02:41.207
It's just saying, OK, roughly,

00:02:41.607 --> 00:02:43.002
if you wanted to construct this

00:02:43.002 --> 00:02:44.433
so it's easier to do the math,

00:02:44.433 --> 00:02:46.433
which the problem writers have done,

00:02:46.433 --> 00:02:50.515
say, look, OK, 30% $0, 25% $1,000.

00:02:50.515 --> 00:02:52.633
This is pretty indicative

00:02:52.633 --> 00:02:55.900
if you had to group all
of the possible costs

00:02:55.900 --> 00:02:58.367
into some major bucket.

00:02:58.367 --> 00:03:00.902
So it's probably at least
a pretty good method

00:03:00.902 --> 00:03:03.700
for figuring out which insurance policy

00:03:03.700 --> 00:03:05.392
someone should use.

00:03:05.653 --> 00:03:07.700
So they say, including
the cost of insurance,

00:03:07.700 --> 00:03:10.367
what are Paul's expected
total medical costs

00:03:10.367 --> 00:03:12.447
with the low-deductible plan?

00:03:12.447 --> 00:03:15.700
Round your answer to the nearest cent.

00:03:15.700 --> 00:03:18.238
Actually, I'll get the
calculator out for this.

00:03:21.683 --> 00:03:24.437
With the low-deductible plan here,

00:03:24.437 --> 00:03:25.833
low-deductible plan,

00:03:25.833 --> 00:03:28.100
he's going to have to
spend his total cost,

00:03:28.100 --> 00:03:30.570
he's going to spend $8,000 no matter what.

00:03:30.831 --> 00:03:32.666
Whoops, what happened to my calculator?

00:03:33.233 --> 00:03:35.568
He's going to spend $8,000 ...

00:03:35.844 --> 00:03:37.365
My God, I'm having issues.

00:03:37.611 --> 00:03:42.366
He's going to spend $8,000 no matter what,

00:03:42.366 --> 00:03:46.433
so that is $8,000.

00:03:46.433 --> 00:03:47.566
Then let's see.

00:03:47.566 --> 00:03:51.233
There is a 30% probability
he spends nothing.

00:03:51.233 --> 00:03:55.234
I could just write that
as plus 0.3 times $0,

00:03:55.234 --> 00:03:56.633
and I will write it just so that you see

00:03:56.633 --> 00:03:58.300
I'm taking that into account.

00:03:58.300 --> 00:04:02.766
There's a 25% chance that he
has $1,000 in medical costs.

00:04:02.766 --> 00:04:04.234
In the low-deductible plan,

00:04:04.234 --> 00:04:06.238
he still has to pay that $1,000,

00:04:06.238 --> 00:04:14.433
so plus 0.25% chance that he pays $1,000,

00:04:14.433 --> 00:04:16.101
$1,000.

00:04:16.101 --> 00:04:18.502
And then you might say,
OK, plus 0.2 times $4,000,

00:04:18.502 --> 00:04:23.633
but remember, if his
medical costs are $4,000,

00:04:23.633 --> 00:04:25.301
he is not going to pay that $4,000.

00:04:25.301 --> 00:04:27.301
He is only going to pay the first $1,000.

00:04:27.301 --> 00:04:30.884
So it's really plus 0.2.

00:04:31.100 --> 00:04:34.708
In this situation, his
out-of-pocket costs are only $1,000.

00:04:34.708 --> 00:04:40.033
The insurance company
will pay the next $3,000.

00:04:40.033 --> 00:04:42.766
So 0.2 times $1,000.

00:04:44.058 --> 00:04:50.433
And then plus 0.2, 20% chance,

00:04:50.433 --> 00:04:52.247
even if he has $7,000 in medical costs,

00:04:52.247 --> 00:04:55.300
he's only going to have
to pay the first $1,000,

00:04:55.300 --> 00:04:58.433
so 0.2 times $1,000 again.

00:04:58.433 --> 00:05:04.366
And then plus 0.05 times,
once again, $1,000.

00:05:04.366 --> 00:05:05.700
If he has $15,000 in expenses,

00:05:05.700 --> 00:05:08.100
he is only going to have
to pay the first $1,000,

00:05:08.100 --> 00:05:10.833
times $1,000.

00:05:10.833 --> 00:05:15.600
And we get $8,700.

00:05:15.634 --> 00:05:16.966
One way you could have thought about it

00:05:16.966 --> 00:05:21.633
is OK, he is going to have
to pay $8,000 no matter what,

00:05:21.633 --> 00:05:25.633
and all of the situations
where he ends up paying,

00:05:25.633 --> 00:05:26.701
that's anything,

00:05:26.701 --> 00:05:28.966
that's these 4 situations right over here,

00:05:28.966 --> 00:05:32.243
there is a 70% probability of falling into

00:05:32.243 --> 00:05:33.833
one of these 4 situations.

00:05:33.833 --> 00:05:37.233
At any one of these, he
only has to pay $1,000.

00:05:37.233 --> 00:05:39.444
The insurance company pays
everything after that.

00:05:39.444 --> 00:05:43.976
So you could say $8,000
plus there is a 70% chance

00:05:43.976 --> 00:05:46.546
that he is going to pay $1,000.

00:05:47.700 --> 00:05:50.768
And once again, this table is
a pretty big simplification

00:05:50.768 --> 00:05:52.368
from the real world.

00:05:52.368 --> 00:05:53.633
There's probably a lot of scenarios

00:05:53.633 --> 00:05:56.300
where you would have to pay $500 or $600

00:05:56.300 --> 00:05:57.568
or whatever it might be.

00:05:57.568 --> 00:05:59.233
But let's just go with this.

00:05:59.233 --> 00:06:01.569
That's essentially a simplification.

00:06:01.569 --> 00:06:05.568
There's a 70% chance that he's
going to have to pay $1,000,

00:06:05.568 --> 00:06:08.433
and so that's $700
expected cost from that,

00:06:08.433 --> 00:06:10.300
plus the $8,000 from the insurance

00:06:10.300 --> 00:06:13.768
gets us to $8,700.

00:06:13.768 --> 00:06:17.700
So let's write that down.

00:06:17.700 --> 00:06:20.966
So $8,700 ...

00:06:20.966 --> 00:06:23.633
My pen is really acting up.

00:06:23.633 --> 00:06:25.010
I don't know what's going on here.

00:06:25.010 --> 00:06:26.967
I think I have to get a new tablet.

00:06:26.967 --> 00:06:28.366
You can't even read that.

00:06:28.366 --> 00:06:30.300
Let me write this.

00:06:30.300 --> 00:06:37.000
$8,700.

00:06:37.633 --> 00:06:39.566
Including the cost of insurance,

00:06:39.566 --> 00:06:42.702
what are Paul's expected
total medical costs

00:06:42.702 --> 00:06:44.233
with the high-deductible plan?

00:06:44.233 --> 00:06:46.533
Round your answer to the nearest cent.

00:06:46.633 --> 00:06:48.900
Let's look at the high-deductible plan.

00:06:48.900 --> 00:06:52.100
He's going to pay $7,500 no matter what,

00:06:52.100 --> 00:06:54.766
$7,500 no matter what,

00:06:54.766 --> 00:06:58.847
and then there's a $0 ...

00:06:58.847 --> 00:07:02.504
We could write $0 times
30% or 30% times $0,

00:07:02.504 --> 00:07:04.100
but that's just going to be $0.

00:07:04.100 --> 00:07:07.366
There is a 25% chance he spends $1,000,

00:07:07.366 --> 00:07:13.100
so plus 0.25 times $1,000.

00:07:13.100 --> 00:07:18.501
There is a 20% chance, plus 0.2,

00:07:18.501 --> 00:07:21.233
he's not going to spend $4,000 here.

00:07:21.233 --> 00:07:23.633
He's going to have to
spend the first $2,500.

00:07:23.633 --> 00:07:26.700
So it's a 20% chance he spends $2,500,

00:07:26.700 --> 00:07:29.700
so times $2,500.

00:07:29.700 --> 00:07:32.100
Insurance company will
pay the next $1,500.

00:07:32.100 --> 00:07:34.567
Plus another 20% chance.

00:07:34.567 --> 00:07:37.833
Even in this situation,
he only has to pay $2,500,

00:07:37.833 --> 00:07:41.633
so times $2,500,

00:07:41.633 --> 00:07:43.166
$2,500.

00:07:43.166 --> 00:07:46.233
And then finally, plus
there's a 5% chance.

00:07:46.233 --> 00:07:49.433
Even in this situation, he only
has to pay the first $2,500,

00:07:49.433 --> 00:07:52.366
times $2,500

00:07:52.366 --> 00:07:57.446
gets us to $8,875.

00:07:57.968 --> 00:08:04.733
So $8,875.

00:08:04.834 --> 00:08:06.700
Once again, you could think about it

00:08:06.700 --> 00:08:10.167
as, OK, there is a 25%
chance that he pays $1,000,

00:08:10.167 --> 00:08:15.131
and then there is a 45%
chance that he pays $2,500.

00:08:15.131 --> 00:08:18.368
All of these situations,
he is paying $2,500.

00:08:18.368 --> 00:08:22.266
But either way, you would get $8,875.

00:08:22.636 --> 00:08:25.366
If Paul wants the best
payoff in the long run

00:08:25.366 --> 00:08:27.582
and must buy 1 of the 2 insurance plans,

00:08:27.582 --> 00:08:29.434
he should purchase the?

00:08:29.434 --> 00:08:34.433
Well, his expected
total cost of insurance,

00:08:34.433 --> 00:08:36.100
including medical costs,

00:08:36.100 --> 00:08:40.566
is lower with the low-deductible plan.

00:08:40.566 --> 00:08:43.167
So this one, he should go with the low,

00:08:43.167 --> 00:08:46.006
low-deductible.

00:08:46.006 --> 00:08:47.100
Which, once again,

00:08:47.100 --> 00:08:50.033
you shouldn't use these
videos as insurance advice.

00:08:50.033 --> 00:08:50.633
This is actually ...

00:08:50.633 --> 00:08:52.256
But also, it's an interesting
way to think about it.

00:08:52.256 --> 00:08:53.633
It's typically ...

00:08:53.633 --> 00:08:55.233
Well, it's not always typically the case

00:08:55.233 --> 00:08:56.767
that the low-deductible
plan is going to have

00:08:56.767 --> 00:08:59.767
a higher long-term ...

00:08:59.767 --> 00:09:03.233
or the low-deductible plan
is going to be a better deal.

00:09:03.233 --> 00:09:04.033
It's usually the ...

00:09:04.033 --> 00:09:06.633
Well, I won't make any
actuarial statements.

00:09:06.633 --> 00:09:08.100
But at least in this situation,

00:09:08.100 --> 00:09:10.567
the low-deductible plan
seems like the better deal.

00:09:10.567 --> 00:09:15.302
He has lower expected total
costs given these probabilities.


Expected value while fishing

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.053
You and your friend Jeremy are fishing in a

00:00:03.053 --> 00:00:06.164
pond that contains ten trout and ten sunfish.

00:00:06.568 --> 00:00:09.505
Each time one of you catches a fish

00:00:09.505 --> 00:00:11.568
you release it back into the water.

00:00:11.803 --> 00:00:14.928
Jeremy offers you the choice of two different bets.

00:00:15.193 --> 00:00:18.786
Bet number one. We don't encourage

00:00:18.786 --> 00:00:20.927
betting but I guess Jeremy wants to bet.

00:00:20.927 --> 00:00:25.477
If the next three fish he catches are all sunfish you will

00:00:25.477 --> 00:00:30.044
pay him 100 dollars, otherwise he will pay you 20 dollars.

00:00:30.334 --> 00:00:34.693
Bet two, if you catch at least two sunfish of the next

00:00:34.693 --> 00:00:39.959
three fish that you catch he will pay you 50 dollars,

00:00:40.177 --> 00:00:44.647
otherwise you will pay him 25 dollars.

00:00:45.021 --> 00:00:50.243
What is the expected value from bet one?

00:00:50.491 --> 00:00:52.915
Round your answer to the nearest cent.

00:00:52.915 --> 00:00:54.146
I encourage you to pause this video

00:00:54.146 --> 00:00:55.824
and try to think about it on your own.

00:00:56.380 --> 00:00:59.458
Let's see. The expected value of bet one.

00:00:59.741 --> 00:01:06.255
The expected value of bet one where we'll say bet one is --

00:01:06.255 --> 00:01:08.772
Let's just define a random variable here

00:01:08.772 --> 00:01:10.646
just to be a little bit better about this.

00:01:10.646 --> 00:01:18.086
Let's say x is equal what you pay, or I guess

00:01:18.086 --> 00:01:19.601
you could say , because you might get something,

00:01:19.601 --> 00:01:28.023
what your profit is from bet one.

00:01:29.969 --> 00:01:31.391
It's a random variable.

00:01:31.625 --> 00:01:39.502
The expected value of x is going to be equal to, let's see.

00:01:39.502 --> 00:01:44.688
What's the probability, it's going to be negative 100

00:01:44.688 --> 00:01:48.750
dollars times the probability that he catches three fish.

00:01:49.187 --> 00:01:58.281
The probably that Jeremy catches three sunfish,

00:01:58.281 --> 00:02:02.718
the next three fish he catches are

00:02:02.718 --> 00:02:05.969
going to be sunfish, times 100 dollars.

00:02:07.522 --> 00:02:09.083
Or I should, well you're going to pay that.

00:02:09.083 --> 00:02:11.647
Since you're paying it we'll put it as negative 100

00:02:11.647 --> 00:02:13.832
because we're saying that this is your expected profit,

00:02:13.832 --> 00:02:15.319
so you're going to lose money there.

00:02:17.010 --> 00:02:20.501
That's going to be one minus this probability,

00:02:20.733 --> 00:02:27.541
the probability that Jeremy catches three sunfish.

00:02:29.002 --> 00:02:32.082
In that situation he'll pay you 20 dollars.

00:02:32.287 --> 00:02:34.002
You get 20 dollars there.

00:02:34.002 --> 00:02:35.941
The important thing is to figure out the

00:02:35.941 --> 00:02:38.613
probability that Jeremy catches three sunfish.

00:02:39.019 --> 00:02:43.768
Well the sunfish are 10 out of the 20 fish so any given

00:02:43.768 --> 00:02:49.221
time he's trying to catch fish there's a 10 in 20 chance,

00:02:49.221 --> 00:02:52.536
or you could say one half probability

00:02:52.536 --> 00:02:54.191
that it's going to be a sunfish.

00:02:54.460 --> 00:02:57.096
The probability that you get three sunfish in a row is

00:02:57.096 --> 00:03:01.165
going to be one half, times one half, times one half.

00:03:01.165 --> 00:03:04.421
They put the fish back in, that's why it stays 10 out

00:03:04.421 --> 00:03:07.046
of the 20 fish. If he wasn't putting the fish back in

00:03:07.046 --> 00:03:10.968
then the second sunfish you would have a nine out of 20

00:03:10.968 --> 00:03:13.860
chance of the second one being a sunfish.

00:03:13.860 --> 00:03:16.397
In this case they keep replacing the

00:03:16.397 --> 00:03:17.824
fish every time they catch it.

00:03:17.824 --> 00:03:21.132
There is a one eighth chance that Jeremy catches

00:03:21.132 --> 00:03:24.834
three sunfish, so this right over here is one eighth.

00:03:25.209 --> 00:03:28.772
And one minus one eighth, this is seven eighths.

00:03:31.010 --> 00:03:33.979
You have a one eighth chance of paying 100 dollars

00:03:33.979 --> 00:03:35.996
and a seven eights chance of getting

00:03:35.996 --> 00:03:38.875
twenty dollars so this gets us to ...

00:03:42.732 --> 00:03:45.562
Your expected profit here,

00:03:45.562 --> 00:03:50.044
there's a one eighths chance, one eighth probability,

00:03:50.044 --> 00:03:54.997
that you lose 100 dollars here, so times negative 100.

00:03:55.504 --> 00:04:00.508
But then there is a seven eighths chance that you get --

00:04:01.261 --> 00:04:03.597
I'll just put parentheses here to make it clear.

00:04:03.597 --> 00:04:05.597
I think the order of operations of the

00:04:05.597 --> 00:04:07.301
calculator would have taken care of it but

00:04:07.301 --> 00:04:09.301
I'll just do it so that it looks the same.

00:04:09.301 --> 00:04:11.472
Seven eighths, there's a seven eighths

00:04:11.472 --> 00:04:14.456
chance that you get 20 dollars.

00:04:15.597 --> 00:04:20.098
Your expected payoff here is positive five dollars.

00:04:20.960 --> 00:04:27.223
Your expected payoff here is equal to five dollars.

00:04:27.540 --> 00:04:30.239
This is your expected value from bet one.

00:04:30.629 --> 00:04:32.223
Now let's think about bet two.

00:04:35.323 --> 00:04:39.086
If you catch at least two sunfish of the next three fish

00:04:39.086 --> 00:04:44.211
you catch he will pay you 50, otherwise you will pay him 25.

00:04:44.725 --> 00:04:48.365
Let's think about the probability of catching at least

00:04:48.365 --> 00:04:53.131
two sunfish of the next three fish that you catch.

00:04:53.574 --> 00:04:56.257
There's a bunch of ways to think about this but since

00:04:56.257 --> 00:05:00.616
there's only three times that you're trying to catch the

00:05:00.616 --> 00:05:02.756
fish and there's only one of two outcomes you could actually

00:05:02.756 --> 00:05:06.304
write all the possible outcomes that are possible here.

00:05:06.648 --> 00:05:11.757
You could get sunfish, sunfish, sunfish.

00:05:12.506 --> 00:05:15.007
You could get, what's the other type of fish that you have?

00:05:15.007 --> 00:05:18.758
Oh, trout. You could have sunfish, sunfish, trout.

00:05:18.758 --> 00:05:22.413
You could have sunfish, trout, sunfish.

00:05:22.413 --> 00:05:25.789
You could have sunfish, trout, trout.

00:05:26.054 --> 00:05:29.476
You could have trout, sunfish, sunfish.

00:05:29.790 --> 00:05:31.975
You could have trout, sunfish, trout.

00:05:32.226 --> 00:05:35.133
You could have trout, trout, sunfish.

00:05:35.746 --> 00:05:38.024
Or you could have all trout.

00:05:38.292 --> 00:05:45.538
You see here that each of these, each time you go there's

00:05:45.771 --> 00:05:48.628
two possibilities, each time you try to catch a fish there's

00:05:48.628 --> 00:05:51.020
two possibilities, so if you're doing it three times

00:05:51.020 --> 00:05:53.113
there's two times two times two possibilities.

00:05:53.113 --> 00:05:55.991
One, two, three, four, five, six,

00:05:55.991 --> 00:05:57.785
seven, eight possibilities here.

00:05:57.785 --> 00:06:01.423
Now out of these eight equally likely possibilities how

00:06:01.423 --> 00:06:05.194
many of them involve you catching at least two sunfish?

00:06:05.799 --> 00:06:09.907
You catch at least two sunfish in this one, in this one,

00:06:09.907 --> 00:06:15.439
in that one, in this one and I think that is it.

00:06:15.819 --> 00:06:18.238
Yep, this is only one sunfish, one sunfish,

00:06:18.238 --> 00:06:21.174
one sunfish and no sunfish.

00:06:21.519 --> 00:06:24.876
In four out of the eight equally likely

00:06:24.876 --> 00:06:27.877
outcomes you catch at least two sunfish.

00:06:28.388 --> 00:06:36.764
Your probability of catching at least two sunfish

00:06:38.075 --> 00:06:42.857
is equal to four eighths or one half.

00:06:43.450 --> 00:06:45.699
Let's see, what's the expected value?

00:06:45.980 --> 00:06:49.219
Let's say Y is the expected profit from bet.

00:06:49.941 --> 00:06:53.692
Let's let Y equals, another random variable,

00:06:53.692 --> 00:07:02.568
is equal to expected profit from bet two.

00:07:02.973 --> 00:07:07.026
The expected value of our random variable Y,

00:07:07.363 --> 00:07:10.911
you have a one half chance that you win.

00:07:10.911 --> 00:07:16.411
You have a one half chance of getting 50 dollars and then

00:07:16.411 --> 00:07:20.024
you have the one half chance, the rest of the probability.

00:07:20.505 --> 00:07:21.770
If there's a one half chance you win

00:07:21.770 --> 00:07:23.761
there's going to be a one minus one half or

00:07:23.761 --> 00:07:26.099
essentially a one half chance that you lose.

00:07:26.556 --> 00:07:31.727
So you have a one half chance of having to pay 25 dollars.

00:07:32.679 --> 00:07:34.316
Let's see what this is.

00:07:34.316 --> 00:07:38.286
This is one half times 50 plus one half times negative 25.

00:07:38.598 --> 00:07:48.708
This is going to be 25 minus 12.50, which is equal to 12.50.

00:07:49.618 --> 00:07:52.926
Your expected value from bet two is 12.50.

00:07:53.692 --> 00:07:55.552
Your friend says he's willing to take

00:07:55.552 --> 00:07:59.616
both bets a combined total of 50 times.

00:08:00.038 --> 00:08:03.004
If you want to maximize your expected value what should you do?

00:08:04.025 --> 00:08:07.021
Well bet number two -- Actually both of them are good bets,

00:08:07.021 --> 00:08:09.801
I guess your friend isn't that sophisticated,

00:08:09.801 --> 00:08:14.740
but bet number two has a higher expected payoff,

00:08:14.740 --> 00:08:17.902
so I would take bet two all of the time.

00:08:18.396 --> 00:08:22.301
I would take bet two all of the time.


Expected profit from lottery ticket

WEBVTT
Kind: captions
Language: en

00:00:00.292 --> 00:00:02.046
Voiceover:Ahmed is playing a lottery game

00:00:02.046 --> 00:00:04.204
where he must pick two
numbers from zero to nine

00:00:04.204 --> 00:00:07.428
and then one letter out of the
26 letter English alphabet.

00:00:07.428 --> 00:00:09.964
He may choose the same number both times.

00:00:09.964 --> 00:00:12.592
If his ticket matches the two numbers

00:00:12.592 --> 00:00:14.265
and one letter drawn in order,

00:00:14.265 --> 00:00:19.879
he wins the grand prize
and receives $10,405.

00:00:19.879 --> 00:00:24.127
If just his letter matches but one or both

00:00:24.127 --> 00:00:25.563
of his numbers do not match,

00:00:25.563 --> 00:00:28.941
he wins the small price of $100.

00:00:28.941 --> 00:00:32.973
Under any other outcome, he
loses and receives nothing.

00:00:32.973 --> 00:00:36.975
The game costs him $5 to play.

00:00:36.975 --> 00:00:40.096
Under any other outcome he
loses and receives nothing.

00:00:40.096 --> 00:00:43.758
He has chosen the ticket 04R.

00:00:43.758 --> 00:00:46.428
Assuming he's paying the $5 to play

00:00:46.428 --> 00:00:48.952
and he picks the ticket 04R.

00:00:50.044 --> 00:00:53.121
Let's say we define a random variable X

00:00:53.121 --> 00:00:55.379
and let's say that this random variable

00:00:55.379 --> 00:01:04.144
is the net profit from
playing this lottery game.

00:01:04.144 --> 00:01:08.156
What is the expected from ...

00:01:08.156 --> 00:01:09.834
I guess we could even say the expected

00:01:09.834 --> 00:01:14.941
from the net profit from playing 04R,

00:01:14.941 --> 00:01:17.333
so Ahmed's particular
ticket right over here.

00:01:17.333 --> 00:01:19.808
Let's just say X is the random variable,

00:01:19.808 --> 00:01:22.400
is the net profit from
playing this ticket.

00:01:22.400 --> 00:01:24.694
What I want to think about in this video

00:01:24.694 --> 00:01:29.525
is what is the expected value of that?

00:01:29.525 --> 00:01:33.527
What is the expected net
profit from playing 04R?

00:01:33.527 --> 00:01:35.362
I encourage you to pause the video

00:01:35.362 --> 00:01:37.037
and think through it on your own.

00:01:37.852 --> 00:01:40.009
Let's think about what expected value is.

00:01:40.009 --> 00:01:42.824
It's the probability of
each of those outcomes

00:01:42.824 --> 00:01:45.531
times the net profit from those outcomes.

00:01:45.531 --> 00:01:48.647
There's the probability
of the grand prize.

00:01:48.647 --> 00:01:51.138
I can write that, let me
do that in that red color.

00:01:51.138 --> 00:01:56.060
There is the probability
of getting the grand prize

00:01:56.998 --> 00:02:00.576
and what would times his net
payoff from the grand prize.

00:02:00.576 --> 00:02:01.512
What would that be?

00:02:01.512 --> 00:02:06.560
Well he gets $10,405 but
that's not his net payoff

00:02:06.560 --> 00:02:08.908
or his net profit I should say.

00:02:08.908 --> 00:02:12.391
His net profit is what he gets
minus what he paid to play.

00:02:12.391 --> 00:02:15.944
He paid $5 to play.

00:02:17.328 --> 00:02:22.307
That's that, plus the probability

00:02:22.307 --> 00:02:28.655
of getting the small
price times the pay off

00:02:28.655 --> 00:02:32.660
of the small price which
is going to be $100

00:02:32.660 --> 00:02:35.170
or times the net profit I guess
if you get the small price.

00:02:35.170 --> 00:02:36.737
You get a payoff of a 100

00:02:36.737 --> 00:02:41.156
minus you have to pay $5 to play

00:02:41.156 --> 00:02:47.354
and then finally you have
the probability of neither.

00:02:47.354 --> 00:02:49.713
You're essentially not winning

00:02:49.713 --> 00:02:54.921
and in that situation,
what is the net profit?

00:02:54.921 --> 00:02:57.471
Well in that situation your
net profit is negative five.

00:02:57.471 --> 00:03:00.033
You paid $5 and you got nothing in return.

00:03:00.033 --> 00:03:01.626
To figure out the expected value,

00:03:01.626 --> 00:03:03.372
you just have to figure
out these probabilities.

00:03:03.372 --> 00:03:05.800
What's the probability of the grand prize?

00:03:05.800 --> 00:03:09.039
I'll do that over here,
probability of grand prize.

00:03:09.039 --> 00:03:12.573
Well the probability that he
gets the first letter right

00:03:12.573 --> 00:03:15.845
is one in 10, there's 10 digits there.

00:03:15.845 --> 00:03:18.405
Probability he gets
the second letter right

00:03:18.405 --> 00:03:20.403
is one in 10, these are all independent

00:03:20.403 --> 00:03:22.086
and probability he gets the letter right,

00:03:22.086 --> 00:03:23.670
there's 26 equally likely letters

00:03:23.670 --> 00:03:25.118
that might be in the actual one

00:03:25.118 --> 00:03:27.848
so he has a one in 26
chance of that one as well.

00:03:27.848 --> 00:03:35.133
The probability of the
grand prize is one in 2600.

00:03:35.133 --> 00:03:38.110
This is one in 2600.

00:03:38.110 --> 00:03:49.228
Now what's the probability
of getting the small price?

00:03:49.228 --> 00:03:53.500
Well let's see, he has a one in 26 chance.

00:03:53.500 --> 00:03:57.375
The small prize is
getting the letter right

00:03:57.375 --> 00:04:00.391
but not getting both of the numbers right.

00:04:01.160 --> 00:04:05.136
He has a one in 26 chance
of getting the letter right

00:04:05.136 --> 00:04:08.567
but we're not done here
just with the one in 26

00:04:08.567 --> 00:04:10.428
because this one in 26,

00:04:10.428 --> 00:04:12.363
this includes all the scenarios

00:04:12.363 --> 00:04:13.939
where he gets the letter right,

00:04:13.939 --> 00:04:16.511
including the scenarios where
he wins the grand prize,

00:04:16.511 --> 00:04:19.235
where he gets the letter and
he gets the two numbers right.

00:04:19.235 --> 00:04:22.288
We need to do is we need to
subtract out the situation,

00:04:22.288 --> 00:04:24.788
the probability of
getting the two numbers,

00:04:24.788 --> 00:04:27.931
getting the letter and
the two numbers right

00:04:27.931 --> 00:04:29.487
and we already know what that is,

00:04:29.487 --> 00:04:31.170
it's one in 2600.

00:04:31.170 --> 00:04:35.343
It's one and 26 minus one and 2600.

00:04:35.343 --> 00:04:37.850
The reason why I have to
subtract out at this 2600

00:04:37.850 --> 00:04:42.445
is he has one in 26 chance
of getting this letter right.

00:04:42.445 --> 00:04:48.988
That includes the scenario
where he gets everything right

00:04:48.988 --> 00:04:52.217
but the small prize is only
where you get the letter

00:04:52.217 --> 00:04:54.710
and one or none of these.

00:04:54.710 --> 00:04:55.842
If you get both of these

00:04:55.842 --> 00:04:57.577
then you're at the grand prize case.

00:04:57.577 --> 00:04:59.745
You essentially have to
subtract out the probability

00:04:59.745 --> 00:05:01.375
that you won the grand prize,

00:05:01.375 --> 00:05:02.866
if you got all three of them

00:05:02.866 --> 00:05:05.240
to figure out the probability
of the small prize.

00:05:05.240 --> 00:05:09.003
Now what's the probability
of essentially losing?

00:05:09.003 --> 00:05:14.107
The probability of neither.

00:05:14.107 --> 00:05:17.777
Well it's just kind of
that's everything else.

00:05:17.777 --> 00:05:21.097
It would be one minus these probabilities

00:05:21.097 --> 00:05:22.269
right over here.

00:05:22.269 --> 00:05:24.779
It would be one minus the probability

00:05:24.779 --> 00:05:25.977
of the small prize.

00:05:25.977 --> 00:05:28.397
The probability of the small

00:05:29.781 --> 00:05:32.428
minus the probability of the grand,

00:05:32.428 --> 00:05:33.835
these are the possible outcomes

00:05:33.835 --> 00:05:36.335
so they have to add up to one or a 100%.

00:05:36.335 --> 00:05:38.971
This is one less probability small

00:05:38.971 --> 00:05:45.872
minus probability of large

00:05:45.872 --> 00:05:50.644
or I'll say grand prize.

00:05:50.644 --> 00:05:51.591
Let's fill this in.

00:05:51.591 --> 00:05:54.345
The probability of the small
one, this right over ...

00:05:54.345 --> 00:05:56.010
I'm using that red too much.

00:05:56.010 --> 00:05:59.553
This right over here is one in 26

00:05:59.553 --> 00:06:04.248
minus one in 2600

00:06:04.248 --> 00:06:09.905
and then this right over
here is one minus the small

00:06:09.905 --> 00:06:14.833
which is one in 26 minus one in 2600

00:06:14.833 --> 00:06:21.616
minus one in 2600.

00:06:21.616 --> 00:06:23.948
This simplifies to let's see,

00:06:23.948 --> 00:06:26.377
this is one minus one over 26

00:06:26.377 --> 00:06:32.928
plus one in 2600 plus
or minus one in 2600.

00:06:32.928 --> 00:06:38.755
These cancel and you're left
with one minus one in 26.

00:06:38.755 --> 00:06:41.028
Why does this make sense?

00:06:41.028 --> 00:06:45.349
The way you get nothing is
if you get the letter wrong.

00:06:45.349 --> 00:06:49.479
You have a one in 26 chance
of getting the letter right

00:06:49.479 --> 00:06:51.359
and then you're going to be
in one of these two categories

00:06:51.359 --> 00:06:53.762
or you have a one minus one 26

00:06:53.762 --> 00:06:56.493
which is equal to 25 of 26.

00:06:56.493 --> 00:07:00.614
You have a 25 26 chance of
getting the letter wrong

00:07:00.614 --> 00:07:02.018
in which case you get nothing,

00:07:02.018 --> 00:07:04.537
in which case you completely lose.

00:07:04.537 --> 00:07:07.480
Let's just get our calculator
out and calculate this

00:07:07.480 --> 00:07:11.593
and we'll round to the nearest penny here.

00:07:11.593 --> 00:07:17.958
Let's see, it is going to be one 2600.

00:07:17.958 --> 00:07:24.370
One divided by 2600 times let's see,

00:07:24.370 --> 00:07:27.684
10,405 minus five is going to be 10,400,

00:07:27.684 --> 00:07:30.977
times 10,400, that's your net profit

00:07:30.977 --> 00:07:33.076
when you win the grand prize

00:07:33.076 --> 00:07:40.004
and then you're going to
have plus one divided by 26

00:07:40.004 --> 00:07:46.059
minus one divided by 2600

00:07:47.520 --> 00:07:51.943
times your net profit for the small price

00:07:51.943 --> 00:07:53.806
is a 100 minus five which is 95,

00:07:53.806 --> 00:08:00.285
and then finally plus 25 26.

00:08:00.285 --> 00:08:06.180
25 divided by 26, actually I'll
put parenthesis around here

00:08:06.180 --> 00:08:07.564
just to make it consistent.

00:08:07.564 --> 00:08:13.197
25 divided by 26 times that net payoff.

00:08:13.197 --> 00:08:15.599
When you got nothing, well
you have to pay out $5

00:08:15.599 --> 00:08:18.525
and you got nothing in
return, times negative five.

00:08:18.525 --> 00:08:21.241
Actually I don't know if
it's going to recognize that

00:08:21.241 --> 00:08:25.732
as times so I'll just
write times negative five

00:08:25.732 --> 00:08:29.493
and let me delete that and
we deserve a drum roll now.

00:08:29.493 --> 00:08:34.264
We get a expected net profit of playing as

00:08:34.264 --> 00:08:41.163
$2.81 if we round up to the nearest penny.

00:08:41.163 --> 00:08:46.481
This is all going to be equal to $2.81.

00:08:46.481 --> 00:08:51.540
This is actually a very
unusual lottery game

00:08:51.540 --> 00:08:55.866
where you have a positive
expected net profit as a player.

00:08:55.866 --> 00:08:57.973
Usually the purpose on
operating the lottery,

00:08:57.973 --> 00:09:00.214
the state, or the casino, whoever it is,

00:09:00.214 --> 00:09:02.437
they're the ones who have
the expected net profit

00:09:02.437 --> 00:09:04.415
and then the player has
the expected net loss

00:09:04.415 --> 00:09:07.024
but this actually would
make rational sense to play

00:09:07.024 --> 00:09:10.651
which is not the case
with most lottery games

00:09:10.651 --> 00:09:16.508
and if by playing you actually
expect a $2.81 net profit.


Getting data from expected value

WEBVTT
Kind: captions
Language: en

00:00:00.368 --> 00:00:03.566
Voiceover:Jamie's dad gave
her a die for her birthday.

00:00:03.566 --> 00:00:05.433
She wanted to make sure it was fair,

00:00:05.433 --> 00:00:09.101
so she took her die to school
and rolled it 500 times

00:00:09.101 --> 00:00:10.833
and kept track of how many times

00:00:10.833 --> 00:00:12.933
the die rolled each number.

00:00:13.033 --> 00:00:15.501
Afterwards, she calculated
the expected value

00:00:15.501 --> 00:00:20.434
of the sum of 20 rolls to be 67.4,

00:00:20.434 --> 00:00:24.967
the expected value of the
sum of 20 rolls to be 67.4.

00:00:24.967 --> 00:00:27.501
On her way home from
school, it was raining,

00:00:27.501 --> 00:00:30.767
and 2 values were washed
away from her data table.

00:00:30.767 --> 00:00:33.679
Find the 2 missing absolute frequencies

00:00:33.679 --> 00:00:35.500
from Jamie's data table.

00:00:35.500 --> 00:00:38.566
So you see here, she
rolled her die 500 times,

00:00:38.566 --> 00:00:40.700
and she wrote down how
many times she got a 2.

00:00:40.700 --> 00:00:44.833
She got a 2 110 times, a
3 95 times, a 4 70 times,

00:00:44.833 --> 00:00:46.766
a 5 75 times,

00:00:46.766 --> 00:00:48.967
and then she had written
down how many times

00:00:48.967 --> 00:00:52.300
she got a 1 and a 6, but
then it got washed away,

00:00:52.300 --> 00:00:55.236
so we need to figure out how
many times she got a 1 and a 6,

00:00:55.236 --> 00:00:57.700
given the information on
this table right over here

00:00:57.700 --> 00:00:59.911
and given the information
that the expected value

00:00:59.911 --> 00:01:03.638
of the sum of 20 rolls is 67.4.

00:01:03.638 --> 00:01:05.100
I encourage you to pause this video

00:01:05.100 --> 00:01:06.567
and think about it on your own

00:01:06.567 --> 00:01:08.840
before I give a go at it.

00:01:09.366 --> 00:01:12.767
So first, let's think about
what this expected value,

00:01:12.767 --> 00:01:16.903
the sum of 20 rolls being 67.4 tells us.

00:01:16.903 --> 00:01:20.034
That means that the
expected value of 1 roll,

00:01:20.034 --> 00:01:23.566
the expected value of the sum of 20 rolls

00:01:23.566 --> 00:01:27.433
is just 20 times the
expected value of 1 roll.

00:01:27.433 --> 00:01:29.646
The expected value of a roll,

00:01:29.646 --> 00:01:30.834
let me do it here,

00:01:30.834 --> 00:01:37.694
expected value of a roll
is going to be equal

00:01:37.694 --> 00:01:42.700
to 67.4 divided by 20.

00:01:42.700 --> 00:01:44.438
We can get our calculator out.

00:01:44.438 --> 00:01:45.433
Let's see.

00:01:45.648 --> 00:01:53.766
So we have 67.4 divided by 20 is 3.37.

00:01:53.766 --> 00:01:57.500
So this is equal to 3.37.

00:01:57.500 --> 00:01:59.266
So how does that help us?

00:01:59.500 --> 00:02:02.633
We know how to calculate an expected value

00:02:02.633 --> 00:02:06.300
given this frequency
table right over here.

00:02:06.300 --> 00:02:08.566
If we say that this
number right over here,

00:02:08.566 --> 00:02:10.567
let's say that's capital A

00:02:10.567 --> 00:02:13.900
and let's say that this
number here is capital B,

00:02:13.900 --> 00:02:17.967
if we were to try to calculate
the expected value of a roll,

00:02:17.967 --> 00:02:20.300
what we really want to do is
take the weighted frequency

00:02:20.300 --> 00:02:21.701
of each of these values,

00:02:21.701 --> 00:02:22.834
the weighted sum.

00:02:22.834 --> 00:02:27.966
So, for example, if we got
a 1 A out of 500 times,

00:02:27.966 --> 00:02:32.966
it would be A out of 500 times 1,

00:02:32.966 --> 00:02:36.365
times 1 plus,

00:02:36.365 --> 00:02:37.967
I'll do this in different colors,

00:02:37.967 --> 00:02:42.434
plus 110 out of 500 times 2,

00:02:42.434 --> 00:02:46.500
plus 110 out of 500 times 2.

00:02:46.500 --> 00:02:47.966
Notice, this is the frequency which was

00:02:47.966 --> 00:02:49.668
they got 2 times 2.

00:02:49.668 --> 00:02:54.033
We're taking a weighted
sum of these values.

00:02:54.033 --> 00:02:59.768
And then plus 95 out of 500 times 3,

00:02:59.768 --> 00:03:05.300
plus 95 out of 500 times 3,

00:03:05.300 --> 00:03:06.900
plus,

00:03:06.900 --> 00:03:08.300
I think you see where this is going,

00:03:08.300 --> 00:03:11.100
70 over 500 times 4,

00:03:11.100 --> 00:03:16.433
plus 70 over 500 times 4,

00:03:16.433 --> 00:03:17.833
almost there,

00:03:17.833 --> 00:03:21.501
plus, let's see, I haven't
used this brown color,

00:03:21.501 --> 00:03:25.101
plus 75 over 500 times,

00:03:25.101 --> 00:03:25.701
I'll do it here,

00:03:25.701 --> 00:03:31.801
plus 75 over 500 times 5.

00:03:31.832 --> 00:03:38.166
Finally, plus B over 500,

00:03:38.166 --> 00:03:45.700
plus B over 500 times 6,

00:03:45.700 --> 00:03:48.500
this is going to give us our
expected value of a roll,

00:03:48.500 --> 00:03:52.034
which is going to be equal to 3.37.

00:03:52.034 --> 00:04:00.266
So all of this is equal to 3.37.

00:04:00.638 --> 00:04:01.835
One thing that we can do,

00:04:01.835 --> 00:04:02.900
since we have all these 500s

00:04:02.900 --> 00:04:04.766
and this denominator right over here,

00:04:04.766 --> 00:04:07.900
let's multiply both sides
of this equation times 500.

00:04:07.900 --> 00:04:11.166
If we do that, the left-hand side becomes,

00:04:11.166 --> 00:04:17.500
well, 500 times A over 500 is
just going to be A plus 110,

00:04:17.500 --> 00:04:21.167
plus 110 times 2.

00:04:21.167 --> 00:04:23.368
So it's going to be 220.

00:04:23.629 --> 00:04:29.998
Plus 95 times 3, that's
going to be 15 less than 300,

00:04:29.998 --> 00:04:32.306
so it's going to be plus 285,

00:04:32.306 --> 00:04:35.500
plus 285,

00:04:35.500 --> 00:04:39.166
and then 70 times 4 is 280,

00:04:39.166 --> 00:04:41.366
plus 280.

00:04:41.366 --> 00:04:46.101
75 times 5 is going to be 350 plus 25,

00:04:46.101 --> 00:04:48.103
375,

00:04:48.103 --> 00:04:50.867
so plus 375,

00:04:52.433 --> 00:04:58.300
plus 6B.

00:04:58.961 --> 00:05:00.766
let me make sure I'm not
skipping any steps here,

00:05:00.766 --> 00:05:06.966
plus 6B is going to be
equal to this times 500,

00:05:06.966 --> 00:05:13.433
and that is going to be
equal to 3.37 times 500

00:05:13.433 --> 00:05:16.235
is equal to 1,685,

00:05:16.235 --> 00:05:18.500
1,685.

00:05:18.500 --> 00:05:20.993
All I did to go from this
step right over here,

00:05:20.993 --> 00:05:23.670
which I set up saying here,
this is the expected value

00:05:23.670 --> 00:05:26.967
of one roll, which we
already know to be 3.37,

00:05:26.967 --> 00:05:31.300
is I just multiplied both
sides of this equation by 500.

00:05:31.300 --> 00:05:33.966
I just did this times 500,

00:05:33.966 --> 00:05:36.833
and I did this times 500,

00:05:36.833 --> 00:05:40.300
and this 500 obviously
cancels with all of these,

00:05:40.300 --> 00:05:43.233
and then 500 times 3.37 is 16.85,

00:05:43.233 --> 00:05:45.034
and so I got this right over here.

00:05:45.034 --> 00:05:47.700
Now, I got 1, 2, 3, 4, 5, 6.

00:05:47.700 --> 00:05:49.034
Yup, I did enough.

00:05:49.034 --> 00:05:50.500
I have the right number of terms.

00:05:50.500 --> 00:05:52.768
I just want to make sure I'm
not making a careless mistake.

00:05:52.768 --> 00:05:54.300
If we want to simplify this,

00:05:54.300 --> 00:05:58.900
we can subtract 220, 285,
280, and 375 from both sides.

00:05:59.668 --> 00:06:02.508
If we did that, we would get A,

00:06:02.508 --> 00:06:04.033
if we subtract that
from the left-hand side,

00:06:04.033 --> 00:06:06.571
we're just going to get A plus 6B,

00:06:06.663 --> 00:06:08.439
A plus 6B.

00:06:08.439 --> 00:06:10.300
And on the right-hand side,

00:06:10.300 --> 00:06:12.034
we are going to get,

00:06:12.034 --> 00:06:13.500
let's get our calculator out,

00:06:13.500 --> 00:06:20.967
1,685 minus 220,

00:06:20.967 --> 00:06:22.367
220,

00:06:22.367 --> 00:06:26.100
minus 285

00:06:26.100 --> 00:06:30.033
minus 280

00:06:30.033 --> 00:06:34.900
minus 375

00:06:34.900 --> 00:06:39.740
gets us to 525.

00:06:39.740 --> 00:06:46.766
So we get A plus 6B is equal to 525.

00:06:46.766 --> 00:06:48.436
You say, "OK, you did all that work,

00:06:48.436 --> 00:06:50.167
"but we still have one
equation with 2 unknowns.

00:06:50.167 --> 00:06:55.534
"How do we figure out what
A and B actually are?"

00:06:55.835 --> 00:06:56.901
We know something else,

00:06:56.901 --> 00:06:58.841
We know, and this is actually
much easier to figure out,

00:06:58.841 --> 00:07:01.767
we know that the sum of this
whole table right over here,

00:07:01.767 --> 00:07:04.966
A plus 110 plus 95 plus 70 plus 75 plus B

00:07:04.966 --> 00:07:06.567
is equal to 500.

00:07:06.567 --> 00:07:07.967
Or if we ...

00:07:07.967 --> 00:07:09.100
Let me write that down.

00:07:09.100 --> 00:07:24.572
So we know that A plus 110
plus 95 plus 70 plus 75 plus B

00:07:24.696 --> 00:07:28.944
needs to be equal to 500.

00:07:28.992 --> 00:07:32.301
Or we could subtract 110
plus 95 plus 70 plus 75

00:07:32.301 --> 00:07:34.688
from both sides and get,

00:07:34.688 --> 00:07:36.502
if you subtract it from
the left-hand side,

00:07:36.502 --> 00:07:39.366
you're just left with A plus B,

00:07:39.366 --> 00:07:41.712
A plus B,

00:07:41.712 --> 00:07:42.834
and on the right-hand side,

00:07:42.834 --> 00:07:46.768
if we start with 500,

00:07:46.768 --> 00:07:50.551
so 500 minus 110

00:07:51.367 --> 00:07:53.290
minus 95

00:07:53.967 --> 00:07:56.293
minus 70

00:07:57.094 --> 00:07:59.489
minus 75

00:07:59.568 --> 00:08:01.986
gets us to 150.

00:08:01.986 --> 00:08:05.034
So A plus B must be equal to 150,

00:08:05.034 --> 00:08:08.567
is equal to 150.

00:08:08.567 --> 00:08:11.566
Now we have a system of 2
equations and 2 unknowns,

00:08:11.566 --> 00:08:13.100
and so we know how to solve those.

00:08:13.100 --> 00:08:15.166
We could do it by substitution

00:08:15.166 --> 00:08:18.300
or we could subtract the
second equation from the first,

00:08:18.300 --> 00:08:19.766
so let's do that.

00:08:19.766 --> 00:08:22.633
Let's subtract the left-hand
side of this equation from that

00:08:22.633 --> 00:08:25.504
or essentially, we could multiply

00:08:25.504 --> 00:08:27.900
this one times a negative 1

00:08:27.900 --> 00:08:30.433
and then add these 2 equations.

00:08:30.433 --> 00:08:32.434
The As are going to cancel out,

00:08:32.434 --> 00:08:34.366
and we are going to be left with

00:08:34.366 --> 00:08:42.100
6B minus B is 5B is equal to 375,

00:08:42.100 --> 00:08:47.100
is equal to 375.

00:08:47.100 --> 00:08:47.900
Did I do that right?

00:08:47.900 --> 00:08:49.905
If I add 125 to this, I get to 500,

00:08:49.905 --> 00:08:52.234
then another 25, I get to 525.

00:08:52.234 --> 00:08:54.767
So 5B is equal to 375,

00:08:54.767 --> 00:08:58.034
or if we divide both sides by 5,

00:08:58.034 --> 00:09:05.134
we get B is equal to 75.

00:09:05.284 --> 00:09:07.433
B is equal to 75.

00:09:07.433 --> 00:09:10.233
This right over here is equal to 75.

00:09:10.233 --> 00:09:13.033
If B is equal to 75, what is A?

00:09:13.033 --> 00:09:16.434
We know that A plus B is equal to 500.

00:09:16.434 --> 00:09:17.833
We figured that out a little while ago

00:09:17.833 --> 00:09:19.368
before we multiplied both sides of this

00:09:19.368 --> 00:09:20.900
times a negative 1.

00:09:20.900 --> 00:09:25.085
We knew that A plus B,

00:09:25.085 --> 00:09:27.433
when B is now 75,

00:09:27.433 --> 00:09:29.833
so we could say A plus 75,

00:09:29.833 --> 00:09:34.771
is equal to 150,

00:09:35.125 --> 00:09:36.700
and that's just from this,

00:09:36.700 --> 00:09:39.633
we figured out that A
plus B is equal to 150

00:09:39.633 --> 00:09:41.900
before we multiplied both
sides times a negative.

00:09:41.900 --> 00:09:43.901
Subtract 75 from both sides,

00:09:43.901 --> 00:09:48.634
you get A is also equal to 75.

00:09:48.634 --> 00:09:51.000
And we are done.


Picking fairly

WEBVTT
Kind: captions
Language: en

00:00:00.615 --> 00:00:02.934
A magician performing at a birthday party

00:00:02.934 --> 00:00:05.351
stands inside a circle of 15 kids.

00:00:05.351 --> 00:00:08.750
He's going to choose a volunteer
and he wants each kid to have

00:00:08.750 --> 00:00:11.518
the same chance of getting
chosen, fair enough.

00:00:11.518 --> 00:00:14.910
Now we have three methods
by which he can do it.

00:00:14.910 --> 00:00:18.487
Let's just think about whether
each of these methods are fair,

00:00:18.487 --> 00:00:21.279
where each kid does have

00:00:21.279 --> 00:00:23.729
the same chance of getting chosen.

00:00:23.729 --> 00:00:26.153
If they don't lead to each kid

00:00:26.153 --> 00:00:27.456
having the same chance of getting chosen,

00:00:27.456 --> 00:00:29.456
think about why that is the case.

00:00:29.456 --> 00:00:32.816
So, method one, the magician starts

00:00:32.816 --> 00:00:35.737
with the birthday boy and moves clockwise,

00:00:35.737 --> 00:00:39.880
passing out 100 pieces of
paper numbered one through 100.

00:00:39.880 --> 00:00:41.544
He cycles around the circle

00:00:41.544 --> 00:00:43.345
until all the pieces are distributed.

00:00:43.345 --> 00:00:45.559
He then uses a random number generator

00:00:45.559 --> 00:00:47.705
to pick an integer, one through 100,

00:00:47.705 --> 00:00:50.503
and chooses the volunteer
with that number.

00:00:50.503 --> 00:00:52.088
So just think about what's happening.

00:00:52.088 --> 00:00:53.489
So there are 15 kids in a circle.

00:00:53.489 --> 00:00:59.639
So there's one, two, three,
four, five, six, seven,

00:00:59.639 --> 00:01:04.583
eight, nine, 10, 11, 12, 13, 14, 15.

00:01:04.583 --> 00:01:06.912
I planned that out amazingly well,

00:01:06.912 --> 00:01:09.197
I didn't think I would be
able to fit exactly 15,

00:01:09.197 --> 00:01:10.088
but it worked out.

00:01:10.088 --> 00:01:11.698
So 15 kids in a circle,
and then he's going

00:01:11.698 --> 00:01:13.226
to hand out pieces of paper.

00:01:13.226 --> 00:01:18.699
So he's going to give
one, two, three, four,

00:01:18.699 --> 00:01:27.029
five, six, seven, eight,
nine, 10, 11, 12, 13, 14, 15.

00:01:27.029 --> 00:01:32.275
This person is going to get
pieces of paper one and 16.

00:01:32.275 --> 00:01:34.378
Now this person is
going to get two and 17.

00:01:34.378 --> 00:01:36.315
You're going to keep going
round and round and round

00:01:36.315 --> 00:01:41.003
until all 100 pieces of paper
are going to get distributed.

00:01:41.003 --> 00:01:49.037
Now, something to think about
is whether every child here

00:01:49.037 --> 00:01:52.163
is going to get the same
number of pieces of paper.

00:01:52.163 --> 00:01:54.053
I encourage you to pause this
video and think about that.

00:01:54.053 --> 00:01:56.556
If we just keep cycling
around all the way to 100,

00:01:56.556 --> 00:02:00.920
does each child get the same
number of pieces of paper?

00:02:00.920 --> 00:02:02.823
Well just think about it.

00:02:02.823 --> 00:02:04.726
In order to get the same
number of pieces of paper,

00:02:04.726 --> 00:02:07.729
100 has to be divisible by 15,

00:02:07.729 --> 00:02:09.793
and we know 100 isn't divisible by 15.

00:02:09.793 --> 00:02:13.838
15 goes into 100 six times.

00:02:15.152 --> 00:02:20.384
Six times 15 is 90, and
you have a remainder of 10.

00:02:20.384 --> 00:02:23.336
So what's going to happen is all 15 kids

00:02:23.336 --> 00:02:25.080
are going to get six pieces of paper,

00:02:25.080 --> 00:02:27.432
and then another 10 of the 15 are going

00:02:27.432 --> 00:02:29.928
to get a seventh piece of paper.

00:02:29.928 --> 00:02:32.920
So they're not all getting an equal number

00:02:32.920 --> 00:02:34.040
of pieces of paper.

00:02:34.040 --> 00:02:37.009
So even though he's
randomly picking an interger

00:02:37.009 --> 00:02:39.049
between one and 100 some
of the students are going

00:02:39.049 --> 00:02:41.234
to have a higher chance
than the other ones.

00:02:41.234 --> 00:02:44.187
The 10 that have seven
pieces of paper are going

00:02:44.187 --> 00:02:46.426
to have a higher chance
than the other five

00:02:46.426 --> 00:02:51.042
who only have six pieces of paper.

00:02:51.042 --> 00:02:55.338
So I would say method one is not fair.

00:02:55.338 --> 00:02:58.094
Let me write this down.

00:02:58.094 --> 00:03:01.758
Not fair.

00:03:01.758 --> 00:03:03.350
Sometimes life isn't fair,

00:03:03.350 --> 00:03:05.470
but in this case it's not
fair where we define fair

00:03:05.470 --> 00:03:07.879
as the same chance of getting chosen.

00:03:07.879 --> 00:03:13.214
That's because they all
have different numbers

00:03:13.214 --> 00:03:14.445
of pieces of paper.

00:03:14.445 --> 00:03:17.854
All of the students are not
equally likely to get picked.

00:03:17.854 --> 00:03:20.046
Let's look at method two.

00:03:20.046 --> 00:03:22.381
The magician starts with the birthday boy

00:03:22.381 --> 00:03:26.766
and moves counterclockwise
passing out 75 pieces

00:03:26.766 --> 00:03:28.598
of paper numbered one through 75.

00:03:28.598 --> 00:03:30.534
He cycles around the circle until

00:03:30.534 --> 00:03:31.918
all the pieces are distributed.

00:03:31.918 --> 00:03:33.862
He then uses a random number generator

00:03:33.862 --> 00:03:36.158
to pick an integer between one through 75

00:03:36.158 --> 00:03:37.894
and chooses the volunteer
with that number.

00:03:37.894 --> 00:03:39.807
So I encourage you to pause this video

00:03:39.807 --> 00:03:41.806
and think about whether
this one, method two,

00:03:41.806 --> 00:03:43.565
whether that one is fair.

00:03:43.565 --> 00:03:46.726
Well method two is the same as method one

00:03:46.726 --> 00:03:49.141
except for instead of
using 100 pieces of paper,

00:03:49.141 --> 00:03:51.459
we're using 75 pieces of paper.

00:03:51.459 --> 00:03:55.350
So think about, is 75 divisible by 15?

00:03:55.350 --> 00:04:01.190
And 75 is, five times 15 is 75.

00:04:01.190 --> 00:04:04.234
So in this situation,
each student is going

00:04:04.234 --> 00:04:05.866
to get five pieces of paper.

00:04:05.866 --> 00:04:10.130
Each gets five pieces of paper.

00:04:10.130 --> 00:04:13.449
So they all have an equally
likely chance of getting picked,

00:04:13.449 --> 00:04:16.114
and he's using a random
number generator to pick them.

00:04:16.114 --> 00:04:18.442
So they all have an equally likely chance.

00:04:18.442 --> 00:04:24.097
I would say method two is, indeed, fair.

00:04:24.097 --> 00:04:26.970
They all have the same
chance of getting chosen.

00:04:26.970 --> 00:04:29.922
Now think about method three.

00:04:31.322 --> 00:04:33.894
The magician starts with the birthday boy

00:04:33.894 --> 00:04:35.693
and moves clockwise, passing out 30

00:04:35.693 --> 00:04:37.750
pieces of paper numbered one through 30.

00:04:37.750 --> 00:04:39.157
So they're all going
to get the same number

00:04:39.157 --> 00:04:40.365
of pieces of paper.

00:04:40.365 --> 00:04:41.926
They're all going to get
two pieces of paper each.

00:04:41.926 --> 00:04:43.348
15 children getting two pieces

00:04:43.348 --> 00:04:45.605
of paper each would be 30 pieces of paper.

00:04:45.605 --> 00:04:47.549
So that looks reasonable so far.

00:04:47.549 --> 00:04:48.974
He cycles around the circle

00:04:48.974 --> 00:04:50.381
until all the pieces are distributed.

00:04:50.381 --> 00:04:52.125
Everyone gets two pieces.

00:04:52.125 --> 00:04:53.341
He gives number one to the birthday boy,

00:04:53.341 --> 00:04:54.749
number two to the next kid and so on,

00:04:54.749 --> 00:04:57.178
so that all seems reasonable.

00:04:57.178 --> 00:04:58.650
Kind of consistent with method two

00:04:58.650 --> 00:05:00.402
except instead of 75 it's 30,

00:05:00.402 --> 00:05:02.394
and obviously 75 was overkill.

00:05:02.394 --> 00:05:04.515
Even here this is overkill,

00:05:04.515 --> 00:05:06.675
he just really needs 15 pieces of paper.

00:05:06.675 --> 00:05:10.658
He then counts the number
of windows in the room

00:05:10.658 --> 00:05:14.490
and chooses the volunteer
with that number.

00:05:14.490 --> 00:05:16.435
So the question here is,

00:05:16.435 --> 00:05:19.395
is the number of windows in the room,

00:05:19.395 --> 00:05:22.851
is it random and is it evenly distributed?

00:05:22.851 --> 00:05:27.387
So maybe, maybe you can make a case

00:05:27.387 --> 00:05:29.231
depending on what building it's in,

00:05:29.231 --> 00:05:32.515
and someone's house, it's somewhat random

00:05:32.515 --> 00:05:36.954
on how many windows that
house happens to have,

00:05:36.954 --> 00:05:39.084
the house that's happening
to host the birthday party,

00:05:39.084 --> 00:05:43.035
but it's not going to
be evenly distributed.

00:05:43.035 --> 00:05:49.234
I don't know, there's probably some,

00:05:49.234 --> 00:05:51.186
if you were to sit and
plot all of the houses

00:05:51.186 --> 00:05:52.250
that had a birthday party,

00:05:52.250 --> 00:05:53.626
you'd probably see that it's more likely

00:05:53.626 --> 00:05:56.986
that they'd have 10
windows than one window,

00:05:56.986 --> 00:05:59.466
and definitely more likely that they have

00:05:59.466 --> 00:06:03.558
10 windows than, let's say, 30 windows,

00:06:03.558 --> 00:06:06.035
or even maybe 15 windows even.

00:06:06.035 --> 00:06:09.952
So it's not going to
be evenly distributed.

00:06:10.398 --> 00:06:16.616
Every house has a somewhat
different number of windows,

00:06:16.616 --> 00:06:19.154
and the house that's
happening to host the party

00:06:19.154 --> 00:06:21.650
seems to be somewhat random,

00:06:21.650 --> 00:06:24.515
but it's not going to be
evenly distributed here.

00:06:24.515 --> 00:06:26.962
I would say it's not a
kind of a really good

00:06:26.962 --> 00:06:30.907
random number generator because
it's not evenly generated.

00:06:30.907 --> 00:06:35.081
So I would say method three is not fair.

00:06:35.681 --> 00:06:38.149
The number of windows is not

00:06:38.149 --> 00:06:41.446
a really good random number generator.

00:06:41.446 --> 00:06:43.574
A good random number
generator, we would want,

00:06:43.574 --> 00:06:45.950
say, number one through
75 where any of these

00:06:45.950 --> 00:06:48.950
have an equally likely
chance of getting picked.

00:06:48.950 --> 00:06:50.949
It's somewhat random,

00:06:50.949 --> 00:06:53.246
the number of windows that building has,

00:06:53.246 --> 00:06:57.246
but they're not all equally likely.


Calculating conditional probability

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:02.900
Voiceover:Rahul's two favorite foods

00:00:02.900 --> 00:00:04.533
are bagels and pizza.

00:00:04.967 --> 00:00:08.633
Let A represent the event that he eats

00:00:08.633 --> 00:00:13.233
a bagel for breakfast and let B represent

00:00:13.233 --> 00:00:15.400
the event that he eats pizza for lunch.

00:00:15.700 --> 00:00:16.666
Fair enough.

00:00:16.900 --> 00:00:19.633
On a randomly selected
day, the probability

00:00:19.633 --> 00:00:22.566
that Rahul will eat a bagel for breakfast,

00:00:22.566 --> 00:00:24.767
probability of A, is .6.

00:00:24.767 --> 00:00:25.933
Let me write that down.

00:00:25.968 --> 00:00:28.966
So the probability that he eats a bagel

00:00:28.966 --> 00:00:32.333
for breakfast is 0.6.

00:00:32.833 --> 00:00:35.766
The probability that he
will eat a pizza for lunch,

00:00:35.766 --> 00:00:37.933
probability of event B ...

00:00:38.433 --> 00:00:40.233
So the probability of ...

00:00:40.233 --> 00:00:41.666
Let me do that in that red color.

00:00:42.330 --> 00:00:45.166
The probability of event B, that he eats

00:00:45.166 --> 00:00:48.733
a pizza for lunch, is 0.5.

00:00:49.433 --> 00:00:52.340
And the conditional
probability, that he eats

00:00:52.340 --> 00:00:55.766
a bagel for breakfast given that he eats

00:00:55.766 --> 00:00:58.500
a pizza for lunch, so
probability of event A

00:00:58.500 --> 00:01:00.500
happening, that he eats
a bagel for breakfast,

00:01:00.500 --> 00:01:03.300
given that he's had a pizza for lunch

00:01:03.300 --> 00:01:05.933
is equal to 0.7, which is interesting.

00:01:06.166 --> 00:01:07.330
So let me write this down.

00:01:07.330 --> 00:01:15.100
The probability of A given,
given that B is true.

00:01:15.100 --> 00:01:22.000
Given B, is not 0.6, it's equal to 0.7.

00:01:22.166 --> 00:01:24.433
And because these two
things are not the same,

00:01:24.433 --> 00:01:27.330
because the probability of A by itself

00:01:27.330 --> 00:01:28.833
is different than the probability of A

00:01:28.833 --> 00:01:32.701
given that B is true, this tells us that

00:01:32.701 --> 00:01:35.133
these two events are not independent.

00:01:35.300 --> 00:01:37.766
That we're dealing with
dependent probability.

00:01:37.766 --> 00:01:38.933
This shows us.

00:01:39.330 --> 00:01:42.330
The fact that B being true has changed

00:01:42.330 --> 00:01:45.566
the probability of A
being true, this tells us

00:01:45.566 --> 00:01:47.667
that A and B are dependent.

00:01:47.968 --> 00:01:49.933
Dependent.

00:01:50.566 --> 00:01:51.900
And so when we start thinking ...

00:01:51.900 --> 00:01:53.433
Well actually let's just, before I start

00:01:53.433 --> 00:01:55.833
going on my little soapbox
about dependent events,

00:01:55.833 --> 00:01:57.166
let's just think about what they actually

00:01:57.166 --> 00:01:58.400
want us to figure out.

00:01:58.500 --> 00:02:02.566
So the probability, the probability of A

00:02:02.566 --> 00:02:04.566
given B is equal to 0.7, that's what

00:02:04.566 --> 00:02:05.867
we wrote right over here.

00:02:05.900 --> 00:02:08.350
Based on this information,
what is the probability

00:02:08.350 --> 00:02:10.133
of B given A?

00:02:10.633 --> 00:02:12.433
So they want us to figure
out the probability

00:02:12.433 --> 00:02:14.933
of B given ...

00:02:15.500 --> 00:02:18.533
Probability of B given A.

00:02:19.330 --> 00:02:21.300
That's what they want us to figure out.

00:02:21.300 --> 00:02:22.566
The conditional probability that Rahul

00:02:22.566 --> 00:02:25.167
eats pizza for lunch, given that he eats

00:02:25.167 --> 00:02:28.300
a bagel for breakfast, rounded to the

00:02:28.300 --> 00:02:29.800
nearest hundredth.

00:02:30.633 --> 00:02:31.966
So how would we think about this?

00:02:31.966 --> 00:02:33.366
And I encourage you to
pause this video before

00:02:33.366 --> 00:02:34.866
I work through it.

00:02:35.633 --> 00:02:37.733
So I'm assuming you've given a go at it.

00:02:37.833 --> 00:02:40.151
So the best way to tackle this is to just

00:02:40.151 --> 00:02:42.279
think about, well, what's
the probability that

00:02:42.279 --> 00:02:46.670
both A and B are going to happen?

00:02:46.766 --> 00:02:49.800
Well, the probably of
A and B happening ...

00:02:49.833 --> 00:02:51.267
And let me do this in a new color.

00:02:51.500 --> 00:02:56.533
The probability of A and B happening.

00:02:57.300 --> 00:02:59.300
A and B.

00:02:59.300 --> 00:03:00.933
I want to stay true to the colors.

00:03:01.100 --> 00:03:02.700
Is equal to ...

00:03:02.700 --> 00:03:05.733
There's a couple of ways
you could write this out.

00:03:06.100 --> 00:03:09.827
This is equivalent to,
this is equivalent to

00:03:09.827 --> 00:03:15.466
the probability, probability of A given B.

00:03:16.167 --> 00:03:21.466
Given B, times the probability of B.

00:03:21.500 --> 00:03:22.734
And hopefully that makes sense.

00:03:22.900 --> 00:03:25.100
The probability that B happens and that

00:03:25.100 --> 00:03:26.900
given that B had happened, the probability

00:03:26.900 --> 00:03:28.200
that A happens.

00:03:28.433 --> 00:03:30.466
And that would also be equal to ...

00:03:30.701 --> 00:03:32.833
So obviously this is A and B is happening,

00:03:32.833 --> 00:03:34.500
or you could do it the other way around.

00:03:34.500 --> 00:03:36.767
You could view it as
the probability that B,

00:03:36.767 --> 00:03:42.133
the probability that B given A happens.

00:03:42.701 --> 00:03:47.600
Given A happens, times
the probability of A.

00:03:48.330 --> 00:03:50.833
Times the probability of A.

00:03:50.833 --> 00:03:51.766
This also makes sense.

00:03:51.766 --> 00:03:54.333
What's the probability that A happened?

00:03:54.367 --> 00:03:55.900
And that, given A happened, what's the

00:03:55.900 --> 00:03:57.670
probability of B?

00:03:57.670 --> 00:03:58.833
You multiply those together, you get the

00:03:58.833 --> 00:04:00.800
probability that both happened.

00:04:01.366 --> 00:04:03.600
So why is this helpful for us?

00:04:03.767 --> 00:04:05.833
Well, we know a lot of this information.

00:04:05.833 --> 00:04:10.660
We know the probability
of A given B is 0.7.

00:04:10.660 --> 00:04:12.700
So let me write that, I'll
scroll down a little bit.

00:04:12.700 --> 00:04:14.467
This is 0.7.

00:04:14.566 --> 00:04:17.933
We know that the probability of B is 0.5.

00:04:18.433 --> 00:04:20.867
So this is 0.5.

00:04:20.967 --> 00:04:23.100
So we know that the probability of A and B

00:04:23.100 --> 00:04:24.800
is the product of these two things.

00:04:24.900 --> 00:04:29.666
That's going to be 0.35.

00:04:29.766 --> 00:04:32.166
Seven times five is 35
or, I guess you could say,

00:04:32.166 --> 00:04:35.333
half of .7 is 0.35.

00:04:35.501 --> 00:04:37.134
.5 of .7.

00:04:37.367 --> 00:04:39.567
And that is going to be
equal to what we need

00:04:39.567 --> 00:04:40.566
to figure out.

00:04:40.566 --> 00:04:43.300
Probability of B given A
times probability of A.

00:04:43.300 --> 00:04:44.600
But we know probability of A.

00:04:44.633 --> 00:04:47.467
We know that that is 0.6.

00:04:47.835 --> 00:04:50.266
We know that this is 0.6.

00:04:50.833 --> 00:04:53.166
So just like that, we've
set up a situation,

00:04:53.166 --> 00:04:55.500
an equation, where we can
solve for the probability

00:04:55.500 --> 00:04:56.733
of B given A.

00:04:57.300 --> 00:04:59.466
The probability of B given A.

00:05:00.433 --> 00:05:02.100
Notice, let me just
rewrite it right over here.

00:05:02.100 --> 00:05:03.667
Actually, I'll write this part first.

00:05:03.900 --> 00:05:12.166
0.6, 0.6 times the
probability of B given A.

00:05:12.166 --> 00:05:13.800
Times that, right over there.

00:05:13.900 --> 00:05:15.166
And I'll just copy and paste it so I don't

00:05:15.166 --> 00:05:16.933
have to keep changing colors.

00:05:17.100 --> 00:05:20.266
That, over there, is equal to 0.35.

00:05:20.566 --> 00:05:23.933
Is equal to 0.35.

00:05:24.366 --> 00:05:25.833
And so to solve for the probability of

00:05:25.833 --> 00:05:29.200
B given A, we can just
divide both sides by 0.6.

00:05:29.766 --> 00:05:34.700
0.6, 0.6 and we get the probability of

00:05:34.700 --> 00:05:37.233
B given A is equal to ...

00:05:37.233 --> 00:05:38.600
Let me get our calculator out.

00:05:38.766 --> 00:05:49.767
So 0.35 divided by, divided by 0.6

00:05:49.767 --> 00:05:52.660
and we deserve a little
bit of a drum roll here,

00:05:52.660 --> 00:05:55.730
is .5833 ...

00:05:55.730 --> 00:05:56.230
It keeps going.

00:05:56.230 --> 00:05:58.200
They tell us to round to
the nearest hundredth.

00:05:58.366 --> 00:06:00.666
So it's 0.58.

00:06:01.100 --> 00:06:07.333
Is approximately, is approximately 0.58.

00:06:07.834 --> 00:06:09.733
So notice, this is equal to 0 ...

00:06:09.767 --> 00:06:14.734
or I'll say approximately equal to 0.58.

00:06:14.900 --> 00:06:17.700
Once again, verifying
that these are dependent.

00:06:17.700 --> 00:06:21.900
The probability that B
happens given A is true,

00:06:21.900 --> 00:06:25.566
is higher than just the probability that B

00:06:25.566 --> 00:06:28.766
by itself, or without
knowing anything else.

00:06:28.766 --> 00:06:31.300
Just the probability of B is lower

00:06:31.300 --> 00:06:33.500
than the probability of
B given that you know,

00:06:33.500 --> 00:06:37.700
given that you know A has happened,

00:06:37.700 --> 00:06:39.933
or event A is true.

00:06:40.433 --> 00:06:41.600
And we're done.


Analyzing dependent probability

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.239
Voiceover:Suppose that
Erika simultaneously rolls

00:00:03.239 --> 00:00:06.435
a 6-sided die and a 4-sided die.

00:00:06.435 --> 00:00:09.833
Let A be the event that she rolls doubles,

00:00:09.833 --> 00:00:10.566
let me write this,

00:00:10.566 --> 00:00:14.633
A be the event that she rolls doubles

00:00:14.633 --> 00:00:20.300
and B be the event that
the 4-sided die is a 4.

00:00:20.300 --> 00:00:24.300
Use the sample space of
possible outcomes below

00:00:24.300 --> 00:00:26.647
to answer each of the following questions.

00:00:26.647 --> 00:00:28.034
Fair enough.

00:00:28.034 --> 00:00:30.700
What is probability of A,

00:00:30.700 --> 00:00:34.070
the probability that Erika rolls doubles?

00:00:34.501 --> 00:00:38.500
Over here, we have our sample
space of possible outcomes.

00:00:38.500 --> 00:00:40.646
Each of these are equally likely,

00:00:40.646 --> 00:00:42.359
and so let's see how
many of them there are.

00:00:42.359 --> 00:00:47.702
There are 1, 2, 3, 4 by 1, 2, 3, 4, 5, 6,

00:00:47.702 --> 00:00:49.633
so there are 24 possible outcomes,

00:00:49.633 --> 00:00:50.567
which makes sense.

00:00:50.567 --> 00:00:53.233
There's 4 possible outcomes
for the 4-sided die

00:00:53.233 --> 00:00:55.700
and 6 possible outcomes
for the 6-sided die,

00:00:55.700 --> 00:00:59.901
so you have a total of 24
equally likely outcomes,

00:00:59.901 --> 00:01:02.100
so probability of,

00:01:02.100 --> 00:01:03.300
let me write it here,

00:01:03.300 --> 00:01:09.633
so probability of A is
going to be the fraction

00:01:09.633 --> 00:01:12.168
of the 24 equally likely outcomes

00:01:12.168 --> 00:01:13.701
that involve event A,

00:01:13.701 --> 00:01:15.301
that she rolls doubles.

00:01:15.301 --> 00:01:16.633
Let's think about that.

00:01:16.633 --> 00:01:20.033
This is she has rolled doubles, 1 and a 1.

00:01:20.033 --> 00:01:22.366
They don't look the same
but they're both 1s.

00:01:22.366 --> 00:01:22.968
Let's see.

00:01:22.968 --> 00:01:26.033
We have a 2 and a 2.

00:01:26.033 --> 00:01:28.700
We have a 3 and a 3,

00:01:28.700 --> 00:01:30.913
and we have a 4 and a 4.

00:01:30.913 --> 00:01:32.701
And it's impossible to have a 5 and a 5

00:01:32.701 --> 00:01:35.301
because the 4-sided die only goes up to 4.

00:01:35.301 --> 00:01:37.368
So there's 4 possibilities,

00:01:37.368 --> 00:01:39.836
4 of the 24 equally likely possibilities

00:01:39.836 --> 00:01:42.112
involve rolling doubles.

00:01:42.112 --> 00:01:44.433
There is a 4/24 probability,

00:01:44.433 --> 00:01:47.433
or if we divide the numerator
to the denominator by 4,

00:01:47.433 --> 00:01:51.700
it is a 1/6 probability that Erika,

00:01:51.700 --> 00:01:55.351
a 1/6 probability that
Erika rolls doubles.

00:01:55.766 --> 00:01:57.836
What is probability of B,

00:01:57.836 --> 00:02:00.902
the probability that
the 4-sided die is a 4?

00:02:00.902 --> 00:02:03.966
The probability of B,

00:02:03.966 --> 00:02:07.033
well, once again, there's 24
equally likely possibilities,

00:02:07.033 --> 00:02:11.033
and how many of them involve
the 4-sided die being a 4?

00:02:11.033 --> 00:02:14.369
You have all of these right over here

00:02:14.369 --> 00:02:18.433
involve a 4-sided die being a 4.

00:02:18.433 --> 00:02:21.500
So this is 1, 2, 3, 4, 5, 6

00:02:21.500 --> 00:02:23.901
of the 24 equally likely possibilities,

00:02:23.901 --> 00:02:27.766
or you could say 1/4 of the
equally likely possibilities

00:02:27.766 --> 00:02:32.033
or the probability is 1/4,

00:02:32.033 --> 00:02:33.766
which makes sense
because probability of B,

00:02:33.766 --> 00:02:36.233
it ignores the 6-sided die,

00:02:36.233 --> 00:02:38.234
and it just says what's the probability

00:02:38.234 --> 00:02:39.701
that the 4-sided die is 4?

00:02:39.701 --> 00:02:41.770
Well, that's 1 one of
the 4 possible outcomes

00:02:41.770 --> 00:02:43.935
for that 4-sided die.

00:02:44.500 --> 00:02:47.501
What is the probability of A given B,

00:02:47.501 --> 00:02:49.566
the probability that Erika rolls doubles

00:02:49.566 --> 00:02:52.633
given that the 4-sided die is a 4?

00:02:52.633 --> 00:02:54.706
Let's just think about this a little bit,

00:02:54.706 --> 00:03:04.235
probability of A given
that B has happened,

00:03:04.235 --> 00:03:06.633
given that B has happened.

00:03:06.633 --> 00:03:08.366
Essentially, we are restricting

00:03:08.366 --> 00:03:10.635
our equally likely possibilities now

00:03:10.635 --> 00:03:13.100
to the situation where B has happened.

00:03:13.100 --> 00:03:16.235
Given B means we're assuming
that B has happened.

00:03:16.235 --> 00:03:19.100
Now, we're restricting our sample space

00:03:19.100 --> 00:03:21.633
of possible outcomes where B has happened

00:03:21.633 --> 00:03:23.767
to this right over here.

00:03:23.767 --> 00:03:30.766
Now there are 1, 2, 3, 4, 5,
6 equally likely outcomes.

00:03:30.766 --> 00:03:34.233
How many of them involve A happening?

00:03:34.233 --> 00:03:37.701
This one right over here
that we had already circled,

00:03:37.701 --> 00:03:40.905
this is the one out of the
6 equally likely outcomes

00:03:40.905 --> 00:03:44.366
that involve doubles,

00:03:44.658 --> 00:03:47.433
so there is a 1/6 probability.

00:03:47.433 --> 00:03:48.500
Now that makes sense.

00:03:48.500 --> 00:03:49.568
Let me just write this down.

00:03:49.568 --> 00:03:52.102
This is 1 over 6.

00:03:52.102 --> 00:03:53.634
Why does this make sense?

00:03:53.634 --> 00:03:57.633
Because with a 4-sided
die we're assuming is a 4.

00:03:57.633 --> 00:03:59.700
So essentially, this
is analogous to saying

00:03:59.700 --> 00:04:00.968
when you roll a 6-sided die,

00:04:00.968 --> 00:04:03.966
what's the probability
that you get a 4 as well,

00:04:03.966 --> 00:04:06.368
because that's the only way
you're going to get doubles,

00:04:06.368 --> 00:04:08.580
given that the 4-sided die is 4.

00:04:08.580 --> 00:04:10.366
And we see that right over here.

00:04:10.366 --> 00:04:12.100
The 6-sided die has to be a 4 as well

00:04:12.100 --> 00:04:13.100
in order for this to be doubles

00:04:13.100 --> 00:04:16.233
because we're assuming it's given that B,

00:04:16.233 --> 00:04:17.633
we're given event B,

00:04:17.633 --> 00:04:21.168
we're restricting our
sample space with event B.

00:04:21.168 --> 00:04:23.701
What is the probability of B given A,

00:04:23.701 --> 00:04:25.566
the probability that the 4-sided die is 4

00:04:25.566 --> 00:04:28.433
given that Erika rolls doubles?

00:04:28.433 --> 00:04:30.168
Let's just think about that a little bit.

00:04:30.168 --> 00:04:35.766
The probability of B given A,

00:04:35.766 --> 00:04:42.477
B given that A is true.

00:04:42.766 --> 00:04:44.575
So what's this going to be?

00:04:45.376 --> 00:04:47.700
This means we're going to
restrict our sample space

00:04:47.700 --> 00:04:54.100
to essentially 4 equally likely
outcomes that A has happened

00:04:54.100 --> 00:04:56.500
so where A is true, I guess I could say.

00:04:56.500 --> 00:04:58.766
So there's 1, 2, 4.

00:04:58.766 --> 00:05:01.834
And how many of them
involve event B being true?

00:05:01.834 --> 00:05:04.767
Well, the only one of these 4 that involve

00:05:04.767 --> 00:05:07.700
event B being true is
this one right over here,

00:05:07.700 --> 00:05:09.234
where we've got our doubles.

00:05:09.234 --> 00:05:15.966
So there is a 1/4 probability
that if we assume,

00:05:15.966 --> 00:05:17.768
given that we've gotten doubles,

00:05:17.768 --> 00:05:22.034
the probability that
the 4-sided die is a 4.

00:05:22.034 --> 00:05:24.766
This is a 1/4 probability,

00:05:24.766 --> 00:05:27.234
and that makes sense.

00:05:27.234 --> 00:05:30.100
If we've got doubles and one
of them is a 4-sided die,

00:05:30.100 --> 00:05:31.834
we either have doubles at 1, doubles at 2,

00:05:31.834 --> 00:05:33.566
doubles at 3, or doubles at 4.

00:05:33.566 --> 00:05:35.234
You see that here, doubles 1, doubles 2,

00:05:35.234 --> 00:05:36.834
doubles 3, doubles 4.

00:05:36.834 --> 00:05:39.300
Well, given that, what's the probability

00:05:39.300 --> 00:05:40.700
that the 4-sided die is 4?

00:05:40.700 --> 00:05:43.233
Well, that means that's
one out of these 4 outcomes

00:05:43.233 --> 00:05:45.813
where it's a double 4 is right over here.

00:05:45.967 --> 00:05:46.786
All right.

00:05:46.971 --> 00:05:49.968
What is the probability of A and B,

00:05:49.968 --> 00:05:52.708
the probability that Erika rolls doubles

00:05:52.708 --> 00:05:57.100
and the second die is 4?

00:05:57.100 --> 00:06:00.633
This means both A and B happened.

00:06:01.339 --> 00:06:02.700
Let's look at this.

00:06:02.700 --> 00:06:03.530
Actually let me write it here.

00:06:03.530 --> 00:06:05.833
Let me do it in a new color.

00:06:05.833 --> 00:06:08.433
The probability of,

00:06:08.433 --> 00:06:11.034
and I'll write "and"
here in a neutral color,

00:06:11.034 --> 00:06:14.500
the probability of A and B.

00:06:14.500 --> 00:06:17.538
Probability of A and B is equal to.

00:06:17.676 --> 00:06:19.633
Well, now, we're looking at once again,

00:06:19.633 --> 00:06:22.633
we have 24 equally likely outcomes.

00:06:22.633 --> 00:06:25.501
We have 24 equally likely outcomes.

00:06:25.501 --> 00:06:28.966
How many of them involve A and B?

00:06:28.966 --> 00:06:31.567
To get A and B, you have to have doubles

00:06:31.567 --> 00:06:35.633
and the 4-sided die needs to be a 4.

00:06:35.633 --> 00:06:37.367
Essentially, you have to have doubles 4.

00:06:37.367 --> 00:06:38.967
Well, there's only one outcome

00:06:38.967 --> 00:06:41.500
out of the 24 equally likley outcomes

00:06:41.500 --> 00:06:44.167
that meets that situation,

00:06:44.167 --> 00:06:45.834
this one right over here,

00:06:45.834 --> 00:06:51.234
so there is a 1/24 probability,

00:06:51.234 --> 00:06:54.879
1/24.

00:06:55.634 --> 00:07:00.833
What is probability of A times
probability of B given A?

00:07:00.833 --> 00:07:03.300
Here, we could just go back to
our numbers right over here.

00:07:03.300 --> 00:07:06.833
Probability of A, that's going to be 1/6.

00:07:06.833 --> 00:07:08.925
Let me do that in a magenta color.

00:07:08.925 --> 00:07:11.966
I like to keep my colors,
be careful about my colors.

00:07:11.966 --> 00:07:17.766
That's 1/6 times the
probability of B given A.

00:07:17.996 --> 00:07:21.366
So the probability of B given
A is 1/4 right over here,

00:07:21.366 --> 00:07:23.833
times 1/4,

00:07:23.833 --> 00:07:26.900
which is, curious enough, 1/24,

00:07:26.900 --> 00:07:29.733
1/24.

00:07:29.935 --> 00:07:33.433
What is probability of B times
probability of A given B?

00:07:33.433 --> 00:07:36.617
Probability of B, we figured out, is 1/4,

00:07:36.955 --> 00:07:38.261
1/4,

00:07:38.302 --> 00:07:41.453
and the probability of A given B is 1/6,

00:07:41.592 --> 00:07:45.492
times 1/6,

00:07:45.492 --> 00:07:49.501
which is equal to 1/24.

00:07:49.501 --> 00:07:51.180
Now does it make sense

00:07:51.180 --> 00:07:53.500
that the probability of A and B is 1/24,

00:07:53.500 --> 00:07:56.833
the probability of A times
probability of B given A is 1/24,

00:07:56.833 --> 00:08:00.367
and the probability of B times
probability of A given B,

00:08:00.367 --> 00:08:01.167
they're all 1/24?

00:08:01.167 --> 00:08:03.666
Is this always going to be the case?

00:08:03.967 --> 00:08:04.833
Well, sure.

00:08:04.833 --> 00:08:07.449
Think about what probability
of A and B means.

00:08:07.449 --> 00:08:09.501
What I mean is that they both happened.

00:08:09.501 --> 00:08:11.033
But that's the same way as saying

00:08:11.033 --> 00:08:14.195
what's the probability of,

00:08:14.195 --> 00:08:16.100
let's just say A is happening.

00:08:16.100 --> 00:08:18.102
Well now, for B and A to happen,

00:08:18.102 --> 00:08:19.511
it's just going to be
that times the probability

00:08:19.511 --> 00:08:22.233
that B is true given that A is true,

00:08:22.233 --> 00:08:25.633
because you could say, well,
are you constraining it.

00:08:25.633 --> 00:08:26.641
We're already multiplying

00:08:26.641 --> 00:08:28.633
by the probability of A being true,

00:08:28.633 --> 00:08:30.039
and now we're multiplying
by the probability

00:08:30.039 --> 00:08:32.100
that B is true given A is true.

00:08:32.100 --> 00:08:34.706
I actually often like to swap these around

00:08:34.706 --> 00:08:37.367
just so it gets a little
bit clearer in my head.

00:08:37.367 --> 00:08:39.100
This one, let's just write it like this,

00:08:39.100 --> 00:08:42.833
the probability of B given A

00:08:42.833 --> 00:08:45.967
times the probability of A.

00:08:45.967 --> 00:08:48.567
This is the probability
that event A is true,

00:08:48.567 --> 00:08:52.633
and this is the probability
that event B is true

00:08:52.633 --> 00:08:55.573
given that we know that A is true.

00:08:55.573 --> 00:08:56.711
It completely makes sense

00:08:56.711 --> 00:08:58.167
that this is going to be the same thing

00:08:58.167 --> 00:09:03.233
as the probability of A and B.

00:09:03.233 --> 00:09:05.367
Clearly, this is a
probability of both of these,

00:09:05.367 --> 00:09:06.967
both A and B happening.

00:09:06.967 --> 00:09:08.434
You can go the other way around.

00:09:08.434 --> 00:09:11.233
The probability of A given B

00:09:11.233 --> 00:09:14.233
times the probability of B,

00:09:14.233 --> 00:09:15.434
that would also be,

00:09:15.434 --> 00:09:16.367
so B ...

00:09:16.367 --> 00:09:18.172
We're saying B needs to be true,

00:09:18.172 --> 00:09:19.900
and that given that B is true,

00:09:19.900 --> 00:09:21.500
that A needs to be true as well,

00:09:21.500 --> 00:09:22.633
so it makes complete sense

00:09:22.633 --> 00:09:24.366
that this is going to be the probability

00:09:24.366 --> 00:09:28.467
of A and B as well.


20 Mississippi River

WEBVTT
Kind: captions
Language: en

00:00:00.332 --> 00:00:02.422
[Reading] Most of the sediment and

00:00:02.422 --> 00:00:04.302
nutrients of the Mississippi River

00:00:04.309 --> 00:00:07.549
no longer reach the coastal wetlands,

00:00:07.549 --> 00:00:10.189
a phenomenon that has adversely

00:00:10.189 --> 00:00:11.889
affected the region's

00:00:11.889 --> 00:00:14.839
ecological balance.[stops reading]

00:00:14.839 --> 00:00:16.659
This feels pretty good, I'm kind of

00:00:16.668 --> 00:00:17.968
in the mood for No Error,

00:00:17.968 --> 00:00:18.978
but let me just double-check it.

00:00:18.978 --> 00:00:21.348
So, 'most of the sediment and nutrients

00:00:21.348 --> 00:00:23.038
of the Mississippi River',

00:00:23.038 --> 00:00:27.288
so that's the subject right over here,

00:00:27.288 --> 00:00:31.548
'no longer reach', yeah, this isn't even

00:00:31.548 --> 00:00:33.798
up for debate, no need to underline this,

00:00:33.798 --> 00:00:35.918
but we have Subject-Verb agreement

00:00:35.918 --> 00:00:37.958
where we're talking about a plural,

00:00:37.958 --> 00:00:40.228
'sediment and nutrients' and they

00:00:40.228 --> 00:00:42.548
'no longer reach' and that makes sense,

00:00:42.548 --> 00:00:46.308
'the coastal wetlands, a phenomenon

00:00:46.308 --> 00:00:50.308
that has adversely affected the region's

00:00:50.308 --> 00:00:54.308
ecological balance.' Yeah, this feels really good.

00:00:54.308 --> 00:00:58.308
I would throw in this as a No error.


13 Inflation rate

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.481
The inflation rate in that country is so high that

00:00:04.481 --> 00:00:08.541
even with adjusted wages, most workers can

00:00:08.541 --> 00:00:15.448
barely pay for food, most workers can barely pay for food and shelter

00:00:15.448 --> 00:00:16.510
and then no error.

00:00:16.510 --> 00:00:17.783
This one actually looks right.

00:00:17.783 --> 00:00:21.687
So, the inflation rate in a country is so high, that

00:00:21.687 --> 00:00:28.562
even with adjusted wages, most workers can barely pay for food and shelter.

00:00:28.562 --> 00:00:29.952
Yeah, that feels right.

00:00:29.952 --> 00:00:31.461
I don't really see any real mistakes.

00:00:31.461 --> 00:00:32.416
It makes complete sense.

00:00:32.416 --> 00:00:38.953
The rate of inflation is so high, that most workers can barely get by.

00:00:38.953 --> 00:00:40.049
Uh, that makes a lot of sense.

00:00:40.049 --> 00:00:43.151
I'll go with no error here.


24 Museum funding

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:00.949
[reading]

00:00:00.949 --> 00:00:03.039
'The museum is submitting proposals to several

00:00:03.039 --> 00:00:07.919
foundations in the hope to gain funds to

00:00:07.919 --> 00:00:11.317
build a tropical butterfly conservatory.

00:00:11.317 --> 00:00:13.697
So we have either an error up here

00:00:13.697 --> 00:00:15.330
or we have no error.

00:00:15.330 --> 00:00:17.057
So the part that, as you can tell,

00:00:17.057 --> 00:00:19.287
even when I read it I had a little bit

00:00:19.287 --> 00:00:21.515
of trouble with, is when we read

00:00:21.515 --> 00:00:23.065
'In the hope TO gain'.

00:00:23.065 --> 00:00:25.195
Normally when you're talking about

00:00:25.205 --> 00:00:31.355
'hope' you would say 'in the hope OF'

00:00:31.355 --> 00:00:34.405
whatever the hope is part of.

00:00:34.406 --> 00:00:40.016
So in this case 'in the hope of GAINING funds'.

00:00:40.016 --> 00:00:41.703
You wouldn't really say

00:00:41.703 --> 00:00:44.069
'In the hope TO gain funds'.

00:00:44.069 --> 00:00:46.069
So this one I would change to

00:00:46.069 --> 00:00:54.539
'In the hope of GAINING funds

00:00:54.545 --> 00:00:58.143
to build a tropical butterfly conservatory.'

00:00:58.143 --> 00:01:02.143
So I'd put the error here in D.

00:01:02.143 --> 00:01:05.943
'in the hope TO gain'

00:01:05.943 --> 00:01:08.200
You're 'in the hope OF' something

00:01:08.200 --> 00:01:10.990
not 'in the hope TO' something.


4 Roaming deer

WEBVTT
Kind: captions
Language: en

00:00:00.621 --> 00:00:03.121
- In areas where deer roam freely,

00:00:03.121 --> 00:00:05.756
residents must dress to protect themselves

00:00:05.756 --> 00:00:08.994
against deer ticks that
might transmit diseases.

00:00:08.994 --> 00:00:10.701
This actually looks pretty good.

00:00:10.701 --> 00:00:14.522
In areas, and we're talking
about what type of areas?

00:00:14.522 --> 00:00:18.617
Well, these are areas
where deer roam freely.

00:00:18.617 --> 00:00:21.461
We're starting with this relative pronoun.

00:00:21.461 --> 00:00:23.281
This is the type of areas.

00:00:23.281 --> 00:00:25.868
Areas where deer roam freely.

00:00:25.868 --> 00:00:26.770
This looks pretty good.

00:00:26.770 --> 00:00:28.684
I like it as is, and so I'll
go with the first choice,

00:00:28.684 --> 00:00:30.741
which is always the as is.

00:00:30.741 --> 00:00:33.462
So no change to the underline.

00:00:33.462 --> 00:00:34.574
But let's just go through the other ones,

00:00:34.574 --> 00:00:36.254
to make sure that they're not as good

00:00:36.254 --> 00:00:38.061
or nowhere near as good.

00:00:38.061 --> 00:00:42.374
So in areas roamed by deer freely,

00:00:42.374 --> 00:00:45.208
residents must dress to
protect themselves, et cetera.

00:00:45.208 --> 00:00:48.984
This one isn't as good, because
this part right over here

00:00:48.984 --> 00:00:50.274
is in the passive voice.

00:00:50.274 --> 00:00:52.213
Instead of saying deer roam freely,

00:00:52.213 --> 00:00:54.813
we're saying roamed by deer freely.

00:00:54.813 --> 00:00:57.301
So it sounds a little bit convoluted.

00:00:57.301 --> 00:00:59.512
Most writers will tell
you the passive voice

00:00:59.512 --> 00:01:02.204
you should avoid it,
unless you really need to,

00:01:02.204 --> 00:01:03.522
unless it really does something special

00:01:03.522 --> 00:01:04.734
to what you're writing.

00:01:04.734 --> 00:01:06.084
And so, I would definitely avoid this

00:01:06.084 --> 00:01:08.627
because of that passive voice.

00:01:08.627 --> 00:01:09.857
So B I'd rule out.

00:01:09.857 --> 00:01:11.061
C has the same problem.

00:01:11.061 --> 00:01:13.661
Areas freely roamed by deer.

00:01:13.661 --> 00:01:14.321
It's a little bit better.

00:01:14.321 --> 00:01:17.160
The adverb freely is
in front of the roamed,

00:01:17.160 --> 00:01:18.273
so they're closer together.

00:01:18.273 --> 00:01:19.400
So how are they roaming?

00:01:19.400 --> 00:01:20.700
They're roaming freely.

00:01:20.700 --> 00:01:22.589
But once again, it's in the passive voice.

00:01:22.589 --> 00:01:24.752
Freely roamed by deer,

00:01:24.752 --> 00:01:27.160
instead of just saying
the deer roam freely.

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So I don't like that one as well.

00:01:28.950 --> 00:01:33.950
Both of these are in the passive voice,

00:01:35.040 --> 00:01:37.460
which in general you should avoid.

00:01:37.460 --> 00:01:41.659
Areas in which there are
deer that roam freely.

00:01:41.659 --> 00:01:44.026
So the in which part is okay.

00:01:44.026 --> 00:01:49.015
You could replace a
where with an in which,

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where in this case the which
is the relative pronoun,

00:01:52.871 --> 00:01:54.986
and we put a preposition
here to talk about,

00:01:54.986 --> 00:01:56.343
well, how do we relate to the area?

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Are we doing it on it?

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Are we doing it in it?

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Where are the deer?

00:01:58.872 --> 00:02:00.238
Well, the deer are in it.

00:02:00.238 --> 00:02:01.580
They're in which.

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So, areas in which deer roam freely.

00:02:03.880 --> 00:02:06.943
So you could say in areas
in which deer roam freely.

00:02:06.943 --> 00:02:10.040
The problem I have with
this one is this part.

00:02:10.040 --> 00:02:11.573
There are deer that roam freely.

00:02:11.573 --> 00:02:14.100
There are deer that roam
freely is much longer than

00:02:14.100 --> 00:02:17.680
deer roam freely, unnecessarily
long, unnecessarily wordy.

00:02:17.680 --> 00:02:19.080
And so, if you can get it across

00:02:19.080 --> 00:02:21.240
in a nice clean way, you should.

00:02:21.240 --> 00:02:22.641
And so, I would rule this one out,

00:02:22.641 --> 00:02:24.300
just because it's wordier

00:02:24.300 --> 00:02:26.807
and a little bit more
complex than it needs to

00:02:26.807 --> 00:02:30.760
relative to our original clause here.

00:02:30.760 --> 00:02:33.540
Areas which deer roam free.

00:02:33.540 --> 00:02:38.536
So the problem here is we have
the relative pronoun which,

00:02:38.536 --> 00:02:40.420
but we're not saying how do we relate.

00:02:40.420 --> 00:02:44.168
How do we relate to those
areas that we're referring to?

00:02:44.168 --> 00:02:47.520
In particular, how do
the deer relate to them?

00:02:47.520 --> 00:02:48.628
Well, the deer are in them.

00:02:48.628 --> 00:02:50.204
You need to put that in which.

00:02:50.204 --> 00:02:52.565
This needs an in right over here.

00:02:52.565 --> 00:02:56.847
Without the in, this isn't
right really, grammatically.

00:02:56.847 --> 00:02:59.647
Areas which deer roam free.

00:02:59.647 --> 00:03:01.967
No, the deer are in the area.

00:03:01.967 --> 00:03:04.644
So I'd cross that one out as well.