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Three Criteria Venn Diagrams
Okay, back to Venn diagrams. For ordinary Venn diagrams that we covered in that earlier lesson, we considered populations with two overlapping groups. That’s a real simplification because in real life, there are many overlapping groups, so we certainly could increase the number of overlapping groups. The test loves to present situations in which members of the population can be placed in three overlapping sets.
It almost never goes above three, but it does like having three overlapping sets. So, for example, at a high school, we could look at the students who A, were studying Italian, B, were in the chorus, or C, were on the baseball team. Those three don’t have much to do with one another. They’re really different skill sets entirely, so we’d expect that an individual student could be in all three of them, a student could be in none of them or a student might be in any combination of them, any two of them, any one of them.
So there are a bunch of different possibilities. In a way, we can think of this as three different questions. We could ask each student, are you in Italian, are you studying Italian? Are you in the chorus? And are you on the baseball team? So three yes/no questions.
Because there are three binary choices, the total number of possible scenarios is 2 to the 3rd equals 8. We’ll talk about this more when we get to the counting section, how we did this calculation. But right now, I’ll just show if we imagine the yes/no answers we could give to those questions, we could have any of these eight scenarios.
And these would be in order, the answers to those three questions. Do you study Italian? Do you play, do you sing in the chorus? And do you play baseball? A very good way to represent these eight scenarios is with a three-way Venn diagram.
So much as a two-way Venn diagram had two overlapping circles, a three-way Venn diagram has three overlapping circles. And notice that here we have eight regions, which are labelled with the letters A through F. So there are all kinds of regions. There are regions where it’s in one circle alone, regions where just two circles overlap, and then there’s the central region where three circles overlap, and of course there’s H.
Those would be the students that aren’t in any of the three activities. So these are the eight choices represented visually. Obviously, most of the information we might be given in the question concerns more than one region on that and we saw a little bit of this earlier when we were talking about simpler Venn diagrams, that often what’s stated in the question, the number given in the question is not an individual region.
It’s a sum of regions. So, for example, if we’re given information like the chorus has 120 members or 40 students sing in both the chorus and the play, sing in the chorus and play baseball. So, the chorus has 120 members. Well, the chorus, that’s B, C, E, and F.
It’s those four regions altogether. And sing in the chorus and play baseball, that’s the overlap of the chorus and baseball team. That is C plus F. So the first piece of information gives us a sum of four regions. The second piece of information gives us a sum of two regions.
And, in fact, most of the information we’re gonna get is like that. Most of the realistic information we’ll get in the question will be the sum of regions. One crucial region that is often given is the central region, the folks who do all three. If the question tells us that some specific number in all three, that is one region we definitely know.
Because that central region is often given in the question, is unambiguous, and is included in most of the other common regions, it’s usually good to start with one of these problems from the middle and work outward. In other words, start from that very central region. All right, so here is a question based on these scenarios.
Pause the video and then we’ll go through this question. Okay, there are a total of 400 students at a school and this school offers a chorus, baseball and Italian. This year, 120 students are in the chorus. All right, so that’s the contents of one whole circle.
40 students are in both chorus and Italian, 45 students are in both chorus and baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many students are in none of the three activities? So, certainly the question is asking for the H region, the region outside of all three circles.
So let’s think about this. We’re given T equals 400. That’s our total. We’re told that 120 students are in chorus, so that means the entire green circle, B plus C plus E plus F, has to equal 120. All those add up to 120.
We’re told that 40 students are in both chorus and Italian. So, chorus and Italian, that would be B plus C. B plus C equals 40. 45 students are in both chorus and baseball. So, chorus and baseball, that’s C plus F, so C plus F equals 45. And 15 students do all three.
So that means that C, the central region, is 15. That’s very important. And now we get this a ha, 220 are either in Italian or baseball. Well, Italian or baseball would be all the regions in either the yellow or the purple circle, so it would be A, B, C, D, F, and G. Those six groups together constitute Italian or baseball and those add up to 220.
So, let’s think about this. If C equals 15 and B plus C equals 40, that means that B equals 25. And it also means that if C is 15 and C plus F is 45, then F must be 30. Well, now we know B, C and F, and we can solve for E. So, we have 25 plus 15 plus E plus 30 equals 120.
We solve for E, we get E equals 50. Okay, let’s put that aside a moment. Now, think about it this way. If we add all the people in Italian or baseball, so that A plus B plus C plus D plus F plus G, so that’s 220 and add E, which is an extra 50, well, that accounts for everyone in all three circles.
So anyone who is in any of these activities or any combination of these activities, they add up to 220. So the only people not included in that group are the people in H, the people we’re looking for. So H must be 400 minus everyone else, or 130 people. That’s how many people are in none of these three groups.
In populations in which each individual can be a member of any three categories, use a three-way Venn diagram. Remember that many of the common categories will involve more than one section of the diagram and work from the central region outward.
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