BAM!
Alright, Mr. Tarrou - back again!
We are going to finish our introduction to probability and start talking about the two
general rules of probability.
You know how we say more means add... but that's only if the events are disjoint.
So the probability of A or B equals the probability of A plus the probability of B. If it is impossible
for those two events to occur together.
Well, if it is possible for them to occur together, we need the general addition rule.
And that says that if the probability of A or B equals the probability of A plus the
probability of B.
But, if I just do A plus B, and those events have a chance of occurring at the same time,
that overlapped area is going to get counted twice.
So, we are going to have to do minus the probability of A and B occurring together.
So, that overlapped area, like I just said, is counted twice.
So, you must subtract that area once.
You know, to cancel out the double count.
And a Venn Diagram , which is about what I am going to set up...as shown on the left...but
not yet.
You would not count this area twice, so the diagram simplifies the mathematics.
Indeed, we are going to be learning these two general probability rules - these will
be given to you on your AP Statistics exam, so you don't need to memorize them, but you
do need to know how to read them.
And know how they work.
But, even the textbook will tell you that we do a lot of calculations with Venn Diagrams
and tree diagrams, like I showed you how to do in the previous video - introducing the
three part video of introducing probabilities.
There is an example there of how to do a tree diagram - that's when you want to find the
probability of sequence of events.
Well, I've got this very generic label up here.
Maybe A is someone drinking soda.
B is someone who drinks tea.
And then here is the probability that they both drink the same drink.
I don't have the room to write up the word problem, or the time, or the space to write
those out.
So just kind of imagine a generic setting.
The probability of A is 40% occurring.
The probability of B has 30% occurring.
And the probability of A and B is 12% chance of occurring.
Now, again, I did not write out an entire word problem for us to read and try to break
down.
That's what you want to do when you read the examples in your textbook and you talk to
your teacher in your class.
But, with this very plainly laid out, we can see we are just going to be 0.4 plus 0.3,
minus the overlapped area of 0.12.
Let me show you how you would set up a Venn Diagram from this notation here.
So to set up a Venn Diagram, you draw a rectangle and that rectangle represents the entire sample
space.
Okay?
So the area, or the probability inside the rectangle is going to be 1.
Now, it is not a density curve, but it is kind of like that.
Where this is the entire sample space.
Inside the sample space, we have a couple of events.
A being 40%, but I am just going to put a circle and I am going to call this circle
A. And there is a possibility of these events occurring at the same time, so I am going
to leave an overlap.
Within that overlap, it is going to represent the probability that both of those events
occur.
So this circle is going to be B. Now, here is the Venn Diagram.
We have event A, event B.
We have an overlapped area, which is going to be A and B occurring together.
Then we have some empty space around those circles, because maybe A and B don't even
occur at all.
If it is a survey and it is about someone drinking coke or tea...maybe someone doesn't
drink either one.
That probability would be found in the outside area of the Venn Diagram.
When you set up a Venn Diagram to help you try and answer probability questions, and
by the way, this is not a sequence of event.
This may be just a survey, asking someone one question.
And then, you can get multiple responses from that one question.
A series, or a sequence of events, or trials - that would be solved or worked out through
the tree diagram, like I talked about in the intro to probabilities video.
When you set up a Venn Diagram, you want to do the overlapped area first.
So the probability that A and B occur together is 12%.
So, 0.12.
And that's the probability they both occur together.
Now, the circle for A, the probability of A occurring - period - regardless of B occurring
or not - The entire circle has to represent 40%.
This is why you want to do the overlapped area first.
Because that entire circle must add up to be .4.
Well this entire circle has 0.12 already represented.
So, 40 minus 10 is 30.
Minus 2 is 28.
So this area of the circle has the probability of 0.28.
This circle, and I will explain how you label that in a second…
This circle, this entire event B..well the entire probability of event B occurring is
30%.
We already have 12% accounted in the overlapped area.
This entire circle, by this notation, has to add up to 0.3. 0.12 means that this outside
area is 0.18.
And so those two add up to 30%.
Now, let's see what is left.
We have 28% plus 12% is 40%.
And now we are up to 50%.
And now we are up to 58% total.
If an entire sample space must add up to the probability of one...
Whatever is remaining must go in the outside of the circles.
And it should be within this rectangle - the Venn Diagram is defined, the boarders are
defined by the rectangle.
That is not me just highlighting these two circles.
So, 58%...outside the circles.
One minus 0.58 is 0.42.
Right?
Let me double check that.
Good!
So now we have probabilities all over this Venn Diagram, and I can answer a lot of questions.
I can make a list of notations and questions here, which I may still do, but I want to
show you how to highlight, or how to label this Venn Diagram.
Inside this part of the circle, event A is occurring, but NOT B. So this is the probability
of A and not B - the uppercase C is representing the compliment.
And again, the compliment of an event is either an event occurs or it does not occur.
B with an exponent of c is the compliment of B, or B not occurring.
We are not inside the circle - we are in circle A, and we are not in circle B.
This, as already noted, the overlapped area, is the probability of A and B. Over in this
circle, the 18%, the 18% probability - that is B occurring and A not.
So that notation would be the probability of not A and B. What is the probability of
B occurring without A occuring?
18%.
What's the probability of A occurring and not B? 28% What is the probability of neither
one of them occuring?
Well, that's 42%.
From setting up this Venn Diagram, I can answer a lot more questions than just what is the
probability of A or B. By the way, I didn't even answer that question.
The probability of A or B...well, this would say, without the Venn Diagram, 0.4 plus 0.3
minus 0.12.
Or, with the Venn Diagram set up, we just count all the areas that are either in A or
B. That is 0.28 plus - I am just working my way across the Venn Diagram - 0.12 plus 0.18.
And that comes out to 58%.
So the probability of A or B occurring is 0.58, but with the Venn Diagram, I can find
the probability that neither one occur, the probability that A occurs and not B, or the
probability of not A and B occurring.
See, these Venn Diagrams not only simplify that math, they also make the answering of
other probability questions from the same setting much, much quicker.
Before I move on to the next topic…
Are A and B independent of each other?
One of your probability rules that you learned in the previous video - the intro to probability
rules - says that the probability of A and B equals the probability of A times the probability
of B IF they are independent.
They gave us the probabilty of A. They gave us the probability of B. And they gave us
the probability that they both occur...and that is 0.12.
If I can take A and B and multiply them...that would be 0.4 times 0.3.
And if I get the same answer as I am told their probability of them occurring together
at the same time, Well...if that formula does actually come out to work, it validates the
given numbers, then those are indeed independent.
If we had been told that the change of them occurring together was 20%, then 20% would
not equal 12%, and then our conclusion would be that they are NOT independent.
So again, we do talk about independent trials, and in this setting, we are talking about
independent outcomes.
Alright!
Second page!
Whoo!
I'm sure this is going to be a two part video...because that took a while.
*nah nah nah nah nah* Alrighty then!
I've got just a few minutes and we will start part two in a second.
Now we are going to revisit an earlier video, an earlier topic, about analyzing the probabilities
between categorical variables in a two-way table.
And we talked about finding row and column percents.
Row and column percents are a form of...or they ARE...conditional probabilities.
But, we did not actually introduce the proper statistical notation of what a conditional
probability looks like.
We just knew when to find a row and column percent.
So we are going to revisit that and then move on to another screen and tie in the actual
proper statistics notation that go along with those conditional probabilities.
Well, conditional probabilities...So here we have a two-way table…
And we are comparing the grade earned for a student and what classes they are generally
taking.
Okay.
We have grades of A,B.C or below - of course nobody in my class is doing below a C. Ha.
Ha.
And we have three topic levels of math, english, and social studies.
Between these grade distributions and these three main subjects, we have a total of 2500
students.
From this two-way table, let's answer some questions.
What percent of the students have A's in general?
No condition given.
Just what percent of A’s.
Well that is 541 divided by 2500.
What percent of math students earned an A?
Now, I don't even want a student being considered unless they are specifically a math student.
So, where is the variable of being a math student?
On the side.
So, that means...what percent of math students earned an A...I only care about the students
who are in math classes...that means that we are setting up a row percent.
It is going to be 112...those math students who got an A, divided by all the math students
of 800.
That is a row percent.
What if I turned the question around, and I said what percent of students with an A
are math students?
So in this sentence the condition is you are a math student.
In this sentence, the condition is that you an A student.
A students is the top of the column.
Instead of doing a row percent for math being a condition, we are doing a column percent
where the grade is the condition.
We have 112 divided by 541.
My last example is what percent earned an A AND was a math student.
Do you see the change in language there?
There is no condition given.
Just what percent of all the students got a A and were in math.
That's 112 out of 2500.
I'm Mr. Tarrou - part one is done, part 2 coming up!
WHOO!