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Venn Diagrams

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Hi, this tutorial covers a type of graph called the Venn diagram. All right, so let's just start with some data. The following data is from a quickie poll and the poll question is about baseball, about the Minnesota Twins.

So should the Twins trade Willingham and Span? So drop Josh Willingham and Dennard Span. And there were four different outcomes. So trade both of them, was yes on Willingham no on Span, yes on Span, no on Willingham, don't trade either of them.

OK, so then we can see the percentage of votes over here, 14%, 4%, 49%, 33%. And it's always good to read the editor's note here. Instant polls are intended as entertainment. They're not considered to be true measurements of public opinion. But it gives us a nice data set to work with. Even though quickie polls are generally just used for entertainment.

All right, so the previous data can be displayed on a Venn diagram. Venn diagrams are graphical displays important to the study of probability. So then, to define a Venn diagram, it's a diagram used to provide a visual way of representing relationships between different sets.

OK, so when you construct a Venn diagram, what I always like to do is start with a rectangle. So this rectangle is going to represent the entire sample space. And then what I'm going to do is I'm going to let-- I'm going to have two circles here which are going to represent two different sets.

So this set is going to be the people in favor of trading Dennard Span. Now, I'm going to draw another circle that represents the people that want that would like to trade Josh Willingham. Now, since there were people that wanted to trade both, what I'm going to do is I'm going to overlap these two circles. And then I'm going to write Willingham above that circle so we understand that the set of those people are the people that want to trade Willingham.

OK, then usually what we do is we put in percentages. So the easiest percentage usually to put in is the overlapping percentage. So from the data, there were 14% wanted to trade both Span and Willingham. So what I'm going to do is I'm going to put 0.14 in the overlapping section here.

Now, what goes here are the people that want a trade Span but not Willingham. So that data was provided for us. That was 49%, so 0.49. The people that wanted to trade Willingham but not Span, that ended up being 0.04.

Now the people that don't want to trade either, so that 33%, 33% is going to go inside of the box but not inside of any of the circles. Now, what's important is that all of the probabilities that you have marked in your Venn diagram need to add up to 1. OK?

So let's just go and make sure that those four numbers all add up to 1. So we have 0.49 plus 0.14 plus point 0.04 plus 0.33. So if we add up all of those numbers, we do end up with 1, which means 100%. So we do have 100% accounted for.

So this is just a good way of displaying probabilities. OK, let's take a look at another example here. And we're going to make another Venn diagram. So 100 people were randomly chosen. 20 had blond hair, 63 had brown eyes, and 9 had both blonde hair and brown eyes.

So if we're going to represent that with a Venn diagram, again, I draw a box here to represent the sample space, rectangle. And I'm going to have one circle represent those with blond hair, one circle represent those with brown eyes. And I am going to overlap them because there were some that were included in both of these two sets.

These circles don't have to overlap. If there were no people that had both, I would draw them as two separate circles with no overlapping region. So again, the easiest number to put in there is the 9. I like to put use them as proportions though, so 9 out of 100 is 0.09.

So that's the percent or the proportion of those with both. And let me make sure I also label these. So I'm going to have this be blond hair and this be brown eyes.

Now we need to be a little bit careful here. The people that are in this region are only those with blond hair, not with brown eyes. So the 20 here, we don't necessarily know what color eyes they have.

But since 9 of them are already accounted for here, then the remaining have to be the 11. So that ends up being 0.11. So 20.2 minus 0.09.

Now with brown eyes, here they're brown eyes but not blond hair. So we need to take the 63 and subtract the 9. So if we subtract 63 minus 9. 54 divide by 100. We end up with 0.54 there.

So that goes here. And then we also need to figure out then which people weren't accounted for. So who has neither blond hair or brown eyes. So what I'm going to do here is I'm going to add up the proportions that I already have on here.

So I'm going to add up 0.11 plus 0.09 plus 0.54. So those were the three that I had accounted for there. I'm going to add those up. And then I'm going to take 1 minus that.

So 100% minus the 74% that are accounted for, and that's 0.26. So I'm going to put the 0.26 out here. So that means that 26% of people have neither blond hair nor brown eyes.

So one of the key things you need to look for in a Venn diagram is whether you can put your numbers directly into the Venn diagram or if you need to make sure that you're not double counting people. All right, so that has been your tutorial on Venn diagrams. Thanks for watching.