Instructor: Hello, students.
Now we are going to problem six B.
So I'm going to read the problem again.
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 more than six consume caffeine and smoke?
So here, more than six means seven and eight
because we have a total of eight participants.
So let's start writing that.
Again we are going to use the binomial distribution.
As I said before, I like to write the formula again,
so it is here,
X successes equal to...
So now basically we have to do this calculation twice,
once for seven and once for eight.
So once we do this,
again, I would like to write this a little bit.
So eight factorial means eight times seven factorial.
In the bottom we have now,
it's seven factorial in the denominator,
and eight minus seven is one,
and then it is one factorial.
So this one would be 0.4 to the power seven.
Again, as you use your calculator,
you will get .00164.
This one will be
0.6 to the power of one.
So once we do this calculation, we get 0.0079.
Okay, so what I'm going to do is I'm going
to put this one actually up there
so I have enough space to do the other calculation
here.
So for eight, we have to kind of repeat the process
except the numbers, of course, will change.
It's going to be eight factorial up here.
It's going to be eight factorial down here,
eight minus eight factorial.
Here it's going to be 0.4 to the power eight.
And again, here it's going to be one minus 0.4,
8 minus eight.
So once we do this algebraically,
what we get as the result here is
0.0007.
So for our final answer, what we have to do is add them.
So going back to the question of this problem,
it says, "What is the probability that more
than six consume caffeine and smoke?"
So that probability is going to be 0.0086,
and, of course, you can multiply it by 100
and get it into the percentage.