This video is provided as supplementary
material for courses taught at Howard Community College, and this is going to be the first of a
series of videos about conditional probability. Conditional probability can be somewhat difficult to understand so what I wanted to do is start with a
couple of fairly simple, intuitive examples and then go on to something
more difficult. So here's the first example. In a group of 100 students, 40 are taking algebra, 30 are taking biology, and 20 are taking both algebra and
biology. If a student chosen at random is taking algebra, what is the probability that he or she
is taking biology? Let's visualize this information with
a Venn diagram. So I've got 100 students all
together. That's going to be the universe. Within the universe are two groups. There's what we'll call group A, for the
students who are taking algebra. We have 40 students in that
group. And then there's, overlapping with that, group B. Group B is for students taking biology, and there are 30 student in that group. And in that overlapping area are
the students who are taking algebra and biology. And there are
20 students there. Now if the question had only asked "what's the probability that any student
chosen at random is taking biology?" In other words, what's the probability of B?, that would be a fairly simple answer to get. There are 30 students altogether
taking biology and there are 100 students total, so the probability of B is 30 over 100, or 30%. But what the question asks you is this,
it says they've already chosen a student at random, and that student is taking algebra. What's the probability that
he or she is taking biology? So this is conditional probability. We're going to write the probability of B given A. Let me explain this notation. The probability, and then in parentheses
I've got B and a bar and an A. The B stands for biology, the probability that a student is taking biology, and the
bar is "given that". In other words, we already know that the student is taking algebra. So we're looking for the probability that
a student istaking biology if we already know that the student is taking algebra. The formula is going to be that the probability of B given A is the probability of the intersection of A and B divided by the probability of A. Now let's put some numbers in here
and this will begin to make sense. So what's the probability of the
intersection of A and B? That intersection is 20 students and
there are 100 students altogether. So the probability of 20 students out of 100 is 20 over 100. And I can write that as 0.2. The probability A... since they were 40 students in group A, 40 students taking
algebra, and 100 students altogether... the probability of A is going to be 40 over 100, or 0.4. And that reduces to 1/2, which is as 50%. Now notice that 50% is different than the 30% that we
got when we didn't worry about whether the student was taking algebra. The fifty percent makes sense if we look
at the Venn diagram. Looking at the Venn diagram what we see is that, of the 40 students who have decided
to take algebra, 20 of those, half of those, have also
decided to take biology. So if we know that a student is taking algebra, we also know that there's a 50% chance that the student is also taking biology,
since half of the algebra students decided to take biology. So this wasn't totally random. It was
the students making up their minds to do this, and it also wasn't something that would be impossible to do
without this formula. We could probably look at the diagram
and say 'oh, it looks like half of the students taking algebra are also taking biology.' So the probability is 50%. So this was the first of a couple of
intuitive problems I wanted to give you. Here's the second. This one is somewhat the same.
In a group of 100 students, 40 were randomly put in a Monday art class. Independent of
that event, 30 students were randomly put
in a Tuesday botany class. If a student chosen at random is in the
art class, what is the probability that he or she
is also in the botany class? So let's do a Venn diagram again. Once again the universe consists
of 100 students. Once again there are two groups. It's also group A and B, but now it's art and botany. So in group A, we have 40 students. And in group B, for botany, we have 30 students. And we don't know what the
overlap area is. These students were chosen at random. It's
not as if the students in the art class decided they would take botany. So we can just figure out what the
probability is that a student is taking both. The probability that a student is in art and botany, the probability of the
intersection of A and B, is just the probability of A times the probability of B, because these are basically two independent events. So the probability of A is going to be the number of students in the art class,
which is 40, over the total number of students, which
is 100. So that's 40 over 100. So the probability of A is 40 over 100, or 0.4. The probability of B is going to be 30, for the number of students in the
botany class, divided by 100, the total number of students. So that's 30 over 100, or 0.3. We multiply those two out (I'll do that down here.) and we get 0.12, or 12%. Let's leave it at 0.12. So now we know that 12% of the students are
in that overlap area. If it's 12% of the students
in there's 100 altogether, we can just put a 12 in here
because there would be 12 students in that area. And now we want to figure out the probability that a student is in the botany class, the probability of B, given that the student is in the art class. So we've already chosen a student. The student is in the art class. What's the probability that he or she is
in botany? So the formula once again is going to be
that the probability equals... That's the probability of B given A equals the probability of the
intersection of A and B divided by the probability of A, the event which already happened. Well we've already figured out that the
probability of the intersection of A and B is 0.12. The probability of A is going to be 0.4, since there are 40
students in group A and 100 students altogether.
That's 40 over 100, or 0.4. Let's write that as 0.40, and then we can think of that as just 12 over 40. And if we reduce 12 over 40, we get 3 over 10, and that's 30%. Now this also makes sense and
is kind of intuitively obvious. The student who are taking botany were just randomly chosen, so it's not like some subset of the students in the art class
decided they were going to be in botany. Any students in this whole group of 100 could be in botany, they were just
randomly chosen for that. 30 students were put into botany
and that was 30% of the total number of students. So this formula gives us something that once again is kind of intuitively
obvious. What I want to do now is go on to another
video for something that is not intuitively
obvious and is where this whole formula begins to make sense and becomes
very useful. So stick around and I'll be back with that one.
