Alright! Part 2 of my introduction to the general probability rules! We just got done
talking about the general addition rule. We just got done talking about how this topic
- the general multiplication rule - ties into the lesson we have already done. Which is
finding the row and column percents of a two-way table. And that is not the only setting where
you have the general multiplication rule being required, but it is a really good tie into
a previous topic. For the general multiplication rule, A and B do not need to be independent.
The notation is the probability of A and B is equal to the probability of A times the
probability of B, given that A has already occurred. This type of conditional probability
was again, addressed in my example in the previous three-part video - introduction to
probabilities - where I talked about the performance of students in my class, as far as their grades
were concerned as the semester averages. Then used that success or performance to base their
probability on whether or not they would pass the AP Exam. Those are not independent events.
They are actually very dependent. So, this notation would have been appropriate for that
type of question. Let's put it into symbols and then I will put it into the example I
just said. The probability of A and B equals the probability of A times the probability
of B given A has already occurred. In my example of class performance and the exam performance…
What is the probability that you do well and you pass the exam? Well, that was the probability
you did well, times the probability that you passed the exam given that you did well. Or,
what's the probability that you did poorly and passed the exam? That would be the probability
that you did poorly times the probability of you passing the exam given that you did
poorly. The tree diagram I did for that example looks something like this… Where it was
how did you do in the class and then how did you do on the test? And if you would like
to watch that intro to probability lesson, please do so. Because I am trying to help
you guys out and I've more to talk about with this two-way table, and not about a tree diagram
that i've already set up. But, if you know the sequence of events that you would like
to find the probability of, you use tree diagrams to solve those probabilities, or they can
help you solve those probabilities. Of course, your performance in the class will determine
your success on the test, and that dependent probability is set up in the tree diagram.
This line as you just heard me say, is read: the probability of B given A has already occurred.
The conditional probability - this conditional probability - this part of the formula IS
the conditional part. You know...How did you do on the exam, given that you did well in
the class? That conditional probability can also be checked - is it really conditional?
It may be that it is independent. If you have enough information, you can check. You know,
is the probability of B equal to the probability of B given that A has occurred? If the probability
of the student passing the exam was the same as the probability that student passed the
exam given that they did well in my class, or poorly, then that would show that the grade
or the passing rate, of the exam, was independent of the success in my class. So, again, if
the probability of B... just B occurring alone… is the same as the probability that B occurs
given that A has already occurred... if those two are the same… Then A occurring had no
effect on B, and thus those are independent events. This notation...this formula...one
version of this is given to you on the AP Exam. This is in my book, and I THINK this
is the one on the exam. I guess I could look… Let's see here… Why yes, it is! So if we
take the formula as it is designed in my book, and I divide both sides by the probability
of A I am going to get the probability of B given A has occurred alone, you'll see this
version of the formula. Now, I actually do prefer this version, so I am glad they give
it to you on the AP Exam. One thing to note...the "given that" part - the condition is the denominator.
So, the condition and the denominator are always going to be the same. You may not realize
that the end of that last video, Part One, but when we did the conditional probabilities
out of the two-way table, we were actually adhering to these formulas. I would like to
try and show that to you in the last few minutes of this video.
*Nah nah nah nah nah* I hope this type is not too small for you to see now… Okay!
Hopefully you can read this. We have this two-way table, and I said in this example:
what probability of math students earned an A? Well the condition is being a math student.
We have the probability of A given that you are in math, and it came out to be 112 divided
by 800. Let me try and show you how that ties in with this conditional probability. The
probability of A given B equals...the probability of B given A equals… The probability, or
the number of...now of course this isn't a probability, but you can see that it is talking
about A and B occurring at the same time. Where did the 112 come from?That was the students
getting an A and being a math student. So that 112, while it is not a probability, it
has something to do with A and B, or A and math occurring at the same time. Whereas the
denominator - the probability of B given at A has occurred equals the probability of both
divided by the probability of A. So, the condition, has a lot to do...everything to do...with
the denominator when you set up any conditional probability. Why did we divide by 800 and
not 2500? Because we are talking about math students. And, their are...*nah nah nah nah
nah*...800 math students. So, 112 divided by 800 is very, very similar and almost exactly
the same as the conditional probability formula. Now, I am not saying EXACTLY the same, because
these are counts and not probabilities, but I wanted to show you the tie in with something
you have already learned, with fancy notation, that you are now seeing in Chapter whatever
it is in your book. In mine it is Chapter 6. What percent of students with an A are
math students? Well, now the condition is A. The condition is you have that A grade.
So the probability of being a math student, given that you earned an A, is again, 112
- because again, we need that number where both of those variables are, A and math. That
is 112. Or, A and B. Right? And then the denominator is 541, because of course, the condition now
is not you being a math student, but you earning an A in the class. The denominator is 541.
That A again was at the top of the column and that is the condition. That is the total
number of As earned. So again, your condition is what sets up the denominator under the
conditional probability formula. I'm Mr. Tarrou!