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(male narrator) So a survey asks 200 people,
what beverage do you drink in the morning?
And offers these choices--
tea only, coffee only, or both coffee and tea.
Now suppose that 20 report tea only,
80 report coffee only, and 40 report both--
could we figure out how many people total
drink tea in the morning?
How many people drink neither tea or coffee?
To do this, uh...we're going to use a Venn diagram,
'cause it gives us a nice way
to sort of picture what's going on.
So we're going to imagine the universal set--
this is everybody.
Remember, there's 200 people in this set.
And we're gonna draw a couple of circles here.
The first circle is going to be for coffee,
and the second circle is going to be for-for tea.
Now we can start introducing the information we know.
We know that 20 people report tea only,
so they would be in this region here--
the part that only includes tea, does not include c.
Eighty people report coffee only.
Forty people report both,
so they will be in the intersection of those two sets.
And notice that altogether, we have 140 people there.
We knew that there was 200 people total,
and so that leaves 60 people outside of those two sets.
Now for most problems,
it doesn't really matter whether the picture
is representative size-wise of the actual sizes.
And so it doesn't really bother me
that this circle is the same size as that one.
We're just using them for representation.
So now we can answer the question.
How many people drink tea in the morning,
is another way of saying, what's the size of set tea?
And now from our picture,
we can see that that's gonna be 40 people, plus 20,
is 60 people who drink tea in the morning.
So how many people drink neither tea or coffee?
So how many people are not in the union of those two sets?
Uh...and...we already figured out
when we were counting up that because there's 200 people,
the combination of the two sets contains 140,
that there must be 60 people who drink neither coffee or tea.
Let's look at another one.
Uh...here we have a survey that asks
which online services have you used in the last month:
Twitter, Facebook, or have you used both?
Um...now the survey results showed
that 40% have used Twitter, uh...70% have used Facebook,
and 20% have used both.
But in this case, the 40% who said they've used Twitter
includes those who have used both.
Uh...a little bit different of a...of a scenario here.
And so here, we've got our-our-our Twitter circle
and our...and our Facebook circle, if you will.
And we know that 20% of people have used both.
We know that 40% have used Twitter,
but we already know that, uh...of those 40,
20, uh...have... have used both,
so that leaves 20% who have just used Twitter.
Um...out of our Facebook folks,
we've got 40% who have used Facebook,
20% have already used both,
and so we have 50 who, um...have only used Facebook.
Now to answer our question here,
how many people have used neither Facebook or Twitter?
What we really need... what we're wondering here
is the number of people who have not used...Facebook or Twitter.
So Facebook or Twitter would be the union of the two,
and we're looking for how many people are not in either.
Now using the Venn diagram,
we can answer that pretty easily.
We've got 50, 60, 70, 80, 90% of people here
in the combination of the two sets,
which leaves 10% outside of those two sets.
Now also, you might be thinking,
well, couldn't we have just added up
the number of users in each set?
Well, yes and no,
because that would add up to 110.
What's goin' on here?
Uh...well, we'd be double-counting
the folks in both sets, so if we were gonna do that,
we would need to subtract out the intersection of the sets
to get our 90% in the union of Sets T and F.
And then the complement to that would be our 10%.