Instructor: Hello students.
Here now we are moving forward to problem six.
So I'm just gonna first read the problem.
"A study is conducted to assess
the impact of caffeine consumption,
smoking, alcohol consumption
and physical activity on cardiovascular disease.
Suppose that 40% of participants
consume caffeine and smoke
if eight participants are evaluated,
what is the probability that exactly
half of them consume caffeine and smoke?"
So here from the problem,
we understand that
we are using discrete data outcomes.
So we are looking for participants
who consume caffeine and smoke.
So here we will have to use
the binomial distribution formula.
As always, I encourage my students
to write down the formula first.
Not only there is always partial credit involved
with just even putting the formula correctly,
but also it helps you understand the process better.
(indistinct) eight successes equal to
the formula N factorial,
X factorial,
X minus N factorial, oops.
Multiplied by P to the power X
multiplied by one minus P
to the power N minus X.
So the first thing I'd like to do here
is kind of see what each of these represent for us.
So X here is going to be four
because our problem stated that,
exactly half and exactly half of eight is four.
Then (indistinct) N is going to be eight.
Again, stated in our formula, sorry
stating our problem statement,
our probability here is 0.4 or 40%,
and one minus P then would be of course 0.6
because probability always adds up to one.
So now I'm going to kind of place all of this
and scoot over a little bit here
and going to see what it's going to look like.
So on the top here,
we will have eight factorial
divided by four factorial,
eight minus four factorial.
Then here it's going to be 0.4
multiplied well not multiplied,
it's to the power four
and then it's going to be one minus 0.4
then eight minus four.
So this is basically what we have
to now solve algebraically.
So here again,
I'm going to put all the numbers.
I'm going to put it here.
So basically what is a factorial?
Factorial is just going
from eight all the way to one
and our book does provide it,
but I will do this as well.
So it's going to be eight times seven
times six times five times four
times three and times two and times one.
And on the bottom it's going to be four factorial here.
So it's going to be one times two
times three times four
and again eight minus four is again four.
So it's gonna be another four factorial here
and multiplication sign.
So one times two times three times four.
So now when you start solving this algebraically,
you will see that this is quite simple
as a lot of these will cancel each other out.
So this will cancel this total part, okay?
Now four times two we know is eight,
So this will cancel this eight.
Then we have three left.
So we will multiply three
and basically three times two is what? Six.
So now we will have a lot of this kind of simplified.
So now we have seven times two is 14
and 14 times five is what? 70.
So I'm just gonna write that here.
Seven times two times five.
And also from this side,
we have 0.4,
0.4 to the power four.
That's going to be 0.026,
and this is going to be 0.1296, okay?
So again, put these down here.
Again, I'm just writing them again
just to make sure we are showing
the entire total work
and we do this multiplication.
What we get is 0.235872,
and that basically when we round it up
and do the percentage, it is 23.6%.
So basically what does that mean?
That means going back
to what our question asked us,
"If eight participants are evaluated,
what is the probability exactly half of them
consume caffeine and smoke?"
Well, that probability is 23.6%.
Now, one thing I want to let you know here
is the type of calculator you'll use.
I know a lot of you
may have those sophisticated scientific calculators
and even though yes,
they're fine, they're great
but please keep in mind
those calculators have these built-in memories
and you have to kind of vacate the memories
before you do new calculations.
The programming is also a little bit more complex.
So I suggest my students
for this class just to use a simple calculator.
Actually, a lot of the calculation that I just did here,
I used my home,
basically my mobile home phone,
it's not my home phone
it's my mobile phone,
but my personal phone.
And I use the calculator that is built-in in that phone.
So that's kind of the simply
approach you can take, okay?
And the simpler the approach sometimes the better
because we really are,
kind of in an introductory level by statistics course.
So there's no reason for us
to use really any of those complex calculators, okay?
So it's just my advice.
Of course, if you know
how to use those calculators,
well, yes you can use them.
A lot of those calculators
will do a lot of the calculations for you.
But you do still have to show me your work
because I need to see that
you are understanding the process.
And if you really do not get the exact answer
we are looking for it.
You show me the process.
You show me that you wrote down the correct formula.
Of course there's going to be partial credit.
So again, it is up to you
what type of calculator you use
but that's my suggestion.
And that's about it.
Now we will go to 6B.