4.1 Empirical Probablitly (and, or, given)

Okay, our last video for 4.1

is going to talk
about empirical probability

and how to create different...

um, conditionals,
looking at a contingency table.

So, to get started,

it says if our sample

is representative...

of the population,
then we can also interpret a percentage

we calculate from the contingency table

as a probability.

And so this is going to--
we're going to go deeper into this idea

in the next section.

That's also known as the likelihood
that something will happen.

All right, since a contingency table

is constructed from data
collected through sampling, right...

All of our, um,

examples have been like,
"A survey says," right.

We're going out there in the field,

we're getting our hands dirty,
we're talking to people.

This is also known as an experiment.

We can call it... an empirical.

And so what that means is

when you are getting probability

from doing hands-on experiments,

like talking to people out there
in the field,

taking data, um,

it is exactly what you're witnessing.

That is known as empirical probability,
all right?

And we'll talk about the difference
between another probability

in the next section.

Um, we call this empirical probability,

or it's also known as experimental...

probability.

I like to think E goes with E, right?

So if I'm doing an experiment,
I'm doing a survey,

whatever I'm doing, and I'm just

stating the data as I witness it,

that's known as empirical.

Now, to write probability,

we have to know a certain,
um, notation.

So it's, uh,

we're going
to use probability notation...

Okay, so what I mean by that is

when you're stating the probability,

we state a big P, okay.

That stands for probability.

And then in parentheses,

we put what we're finding.

So this, if I were going to read this,

it says "P of,"

so it's like that function notation,

if you can remember from Math 95.

Um, but really I would say
this is the probability of

and whatever we'd be finding, right?

The probability that people like coffee.

Um, this is just a shorthand
to write it.

Now, either this
will produce a fraction...

Or a decimal.

And this is important.

Just be careful

and pay attention...

to directions.

So MyOpenMath,
they go back and forth on you,

so make sure you're reading
directions carefully.

Sometimes they'll say, tell me--

tell me a decimal
to the third rounding place.

So make sure you're just rounding
where they say

because it does change

throughout the lesson.

All right.

There are three things I want to cover,

and then we'll do an example.

This is an "and" probability,

and that means when it's just
like your truth tables

when the things happen at the same time.

In other words,
to compare it to the truth tables,

is they have to both be true

in order to be true, okay?

So the "and"
is where they are both happening.

Where the "or," again,

just like your truth tables,
it's inclusive,

and what that means is, refresher,

is that either it-- one is happening,

or the other one is happening,

or both of them are happening, okay?

This is something you want
to be very careful

with over counting.

That's a big mistake we make.

Now "given"

is the probability
that a characteristic is present

knowing that another characteristic
is present.

And so what I mean by that is...

The-- so again, I'm going to read this

as not "P-T given coffee," okay.

You're going to read that
as the probability of T given coffee.

This is the probability
that a participant likes tea

given that we already know they
are a coffee drinker,

and so what that means is these

are the denominator...

Is the grand total.

We're-- with-- given your denominator

is what the "given" part is.

This will make more sense
once we actually do an example.

Uh, is... okay.

Okay, so let's kind of tuck that
away in your memory

or your brain,
maybe you don't have memorized yet.

That seems a little unfair.

Um, and let's put it to use.

So our last example

says use the contingency table

for the results of the beverage survey

to find the following.

First,
find the missing value in the table.

So this is similar to MyOpenMath.

They might give you a table
that's not complete.

Well you can find this missing piece

by either realizing,
well, 60 minus the 20

gives me 40

or 95 minus 55 gives me 40.

So, it should give you the same result.

Okay, now we can start, so it says

what is the probability and percent

that someone likes tea?

So I'm going to look in my tea row,

and it looks
like the total amount of people

that like tea are 60.

So again,
I'm using notation probability.

And maybe I can say, "tea,"
or I say "likes tea."

However much information
you want to give

is up to you.

Well it's going to be 60, and remember,

it's out of the grand total.

So that is it as a probability.

Now that's like saying if they wanted it
as a fraction.

Now, if the instructions
say they want it as a decimal,

in my calculator I'm going
to do 60 divided by 150

and I get 0.4.

Well, I know how to convert
that to a percentage.

I hope you all do, too.

And so it would be 40% of people...

like tea.

So again, it all depends
on how the question's being asked.

So if they want probability,
I would list this.

If they say I want a decimal,
I'd list this.

And if it said I wanted percentage,

I'd move the decimal place two places.

All right, let me get rid of this
so it doesn't confuse us.

Now it says what is the probability

that someone does not like coffee?

So I'm going to look
in my "no coffee" column.

So probability...

doesn't like coffee.

Okay, well that'd be 55
out of your grand total.

I didn't ask for percentage,
I didn't ask for a decimal,

so I am fine,
I can just leave it like that.

C, what is the probability in percent

that someone likes coffee?

Okay, here's this piece,
"and"-- I read it backwards.

Likes tea *and* coffee.

So I'm going to say probability tea...

and coffee.

So I need to find where these overlap,

so where to tea and coffee intersect?

Right here.

So, that's the only place,

because remember,
they both have to be true.

So that'd be 40 out of your grand total,

which is 150.

Now, if I wanted this as a percent,

I'm going to go to my calculator
and do 40 divided by 150.

I'm going to have to round this,

so I'm going to use rounding symbols.

And I'm going to say 0.27.

And as a percentage, 27% of people...

like tea and coffee.

The other thing I want to mention
is when you do your write ups

and your work for your exams,

you want to make sure you're
stating your answers like so, okay?

Make sure you're good with notation,

that's a big piece of mathematics.

D, what is the probability

that someone likes coffee?

Ooh, here's this fun one.

Given the--
gosh, I keep doing that, sorry.

Likes tea given they like coffee.

So I'm going to say probability tea...

given coffee.

So you can write it like that,

I just want to show you another symbol.

Probability tea, it's like this long...

slash, that's a shorthand.

I'll never test you on that,
but they mean the same thing.

Given is just this long line
and probability.

So this is what I was saying earlier.

This is now your denominator, okay?

So I'm going to look at my total for coffee.

Here's the coffee row--
or, forgive me, column.

Well, there's 95 people total

that like coffee.

Out of those people, though,

who likes tea?

Oh, it's just these 40.

So it's like this
is making you narrow in, okay?

So the given is no longer looking
at everybody.

It's narrowing in
what you're looking at.

So-- and it didn't ask for percent,

it didn't ask for a decimal,
so I can leave it like that.

E, what is the probability...

someone likes coffee--

so, order does matter,

and I almost messed it up
on that last one--

coffee given they like tea.

So what that's telling me
is I just want to focus

on my tea area.

So here's my tea area.

How many people like tea?

60...

Out of those people,
how many like coffee?

40.

Now I can say,

or I should say,
you can leave it like this.

You could also reduce this
if you wanted to.

There is something nice
of leaving it like this,

because then you clearly see
how many people we're talking about.

Um, again, just pay attention
to the instructions, okay?

So it might say reduce your fraction,

and it most likely won't, though,

because, probability, we like
to see the full picture.

Okay, so the last little example
I have to show you

is the "or" piece,

and this
is where the issue of over-counting

can sometimes happen.

What is the probability someone
likes tea

or coffee?

Okay, so I'm going to go to my chart.

There was no given piece,

so I don't need to focus somewhere,
I just--

my denominator will be all of it.

So I know my denominator is 150.

Now tea or coffee.

So here, they like both, right.

So remember "or,"
it's either they like one

or the other, or both.

Here, they like both.

Here, they like no coffee

but they only like tea.

And here, they like coffee and no tea.

This does nothing for me

because this means they like neither.

So I don't want to list that.

So what I'm going to do
with those three things is add them.

So I have 40...

plus... 20,

plus 55.

If I add those all up,
I get 115 out of 150.

Now here is the thing:
if you're like, wait,

I wanted to do it a different way.

You do have that option,

it just creates some problems

because some of us want to say, okay,

well, 60 people like tea

and 95 people like coffee,

so that should give me everything,

and if I do that though, I get 155,

which is very different than 115.

The problem is you have

counted 40 here

and here, so you've over-counted.

So if you want to do it this way,

then you have to also take away 40,

because you counted it twice.

That is a very common mistake,

so I really want to encourage you

to just look at individual cells.

So just say okay,
here I want to add this one,

I want to add this one, and this one.

So you don't over-count.

Okay, let me know
if you have any questions

and I'll see you in the next video.