- [Instructor] This short screencast
is going to explain the multiplication rule.
Let's consider the following,
we have two events that are not mutually exclusive,
they have some sort of intersection,
event A on the left, event B on the right.
Now, if we look at this,
we have the intersection of A with B.
We can relate this to conditional probability,
so the conditional probability, given that were B,
so if I wrote probability of A,
given that event B has occurred.
Because I have this here as the second,
given that B has occurred,
I can basically pair this down to being on the right.
So in the right circle here,
this event, B, has occurred,
and what we're trying to find is,
what's the probability that A has also occurred?
If you take a look at this, and if you kinda pair this down,
you can see that this problem is essentially as shown here,
where we're only considering that B has happened,
and so it's really, if you take a look at this graphically,
the probability that A has occurred,
given that B has occurred,
is the probability that both have occurred,
the intersection divided by the probability of B.
So that's one way we can write this.
Another way of looking at this is sort of the same way,
but now we're looking at,
what's the probability that B has occurred,
given that A has occurred?
We can just look at the left circle here.
So now, given that A has occurred,
we're talking about just this circle,
so that's why I erased the right side,
but that's gonna be equal
to the probability of the intersection,
divided by the probability that A has occurred, right?
So we can write either one for this example.
Now, look what happens if I isolate the intersection
that's in the numerator on both the right-hand sides,
if I isolate those and I set these equal to each other.
In other words, if I bring P(B) over,
if I bring P(A) over,
and then I set those equal to each other,
what we end up with is the following,
and this is known as the multiplication rule.
Incidentally, these are both equal to,
we could say these are both equal to
the probability of the intersection,
so sometimes that's valuable.
So maybe sometimes you're trying to find the intersection
and you have these other things.
So let's go through an example here.
You work for a pharmaceutical company
that produces a drug called Curesol.
Unfortunately, 10% of the weekly batches are defective.
It's also known that 20% of batches are created
on each day of the five day work week.
Quality control determines that 30%
of all batches produced on Fridays are defective.
If there are 50 defective batches produced
by the company each week,
how many of them were produced on Friday?
So working backwards, in order to determine
how many of them were produced on Friday
of the 50 defective batches,
we need to somehow determine, what's the probability
that a defective batch was produced on Friday?
The first thing to always do in one of these problems
is to define the events.
So I'm gonna define event A as being a defective batch,
so the batch is defective.
Event B is going to be the batch was produced on Friday.
Now let's start to write down the things that we know.
We know that 10% of the weekly batches are defective,
so that means the probability that a batch is defective
is equal to 10%.
We also know that 20% of batches
are made on each day of the work week.
So the probability that a batch
was produced on Friday is 20%.
We also know that 30% of all batches
produced on Fridays are defective.
Given that we have produced a batch on Friday,
it is defective.
We can use that to write out the probability
that a batch is defective,
A, given that it was produced on Friday,
is equal to 30%.
Now we can use the multiplication rule.
The multiplication rule is shown here.
Let's put a check mark next to everything we know.
We know probability of A, we know probability of B,
and we know probability of A, given that B has occurred,
and now this is just a matter of rearranging
and solving for the unknown,
which is the probability of B, given A.
So given that a batch is defective,
what's the probability that it was produced on Friday?
Knowing that, we can just multiply
by our 50 defective batches
that are produced by the company,
and that will give us the number of batches
of those defective ones that were produced on Friday.
So when we do this,
when we put these numbers into that equation,
we end up with the probably of B
being produced on Friday, given that A has occurred,
the batch was defective, is equal to 60%,
that means that 60% of all batches that are defective
were produced on Friday.
And so if we multiply 60% by 50,
we end up with 30 batches
of those 50 defective batches were produced on Friday.
So hopefully you learned a little bit more
about the multiplication rule in this screencast.
Thanks for watching.