Two-way frequency tables and Venn diagrams

- So what I have here are 12 pieces of candy. The ones that are colored in brown have chocolate on the outside and the ones that have a C on them means that they have coconut on the inside. So for example, this one over here in the top left, it's made out of chocolate on the outside, but it doesn't have coconut on the inside. While this one right over here is chocolate on the outside and it has coconut on the inside. While this one, whoops, I didn't want to do that, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. This one right over here has neither chocolate nor coconut. What I want to think about is ways to represent this information that we are looking at. One way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. The way it's typically done, the convention, is that you would make a rectangle to represent the universe that you care about, in this case it would be all the chocolates. All the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here. Then I'll draw circles to represent the sets that I care about. So say for this one I care about the set of the things that have chocolate. So I'll draw that with a circle. You could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set, chocolate. That is my chocolate set. Then I want to have a coconut set. So coconut. Once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. Now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see, we have one, two, three, four, five, six have chocolate but no coconut. Let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. Once again, I'm not talking about the whole brown thing, I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Chocolate and coconut. Well that's going to be one, two, three. So three of them have chocolate and coconut. Notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well six plus three, nine. How many total have coconut? Well we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well there's only one with coconut and no chocolate. So that's that one right over there, and that represents this are that I'm shading in in white. So how many total coconut are there? Well one plus three, or four, and you see that. One, two, three, four. The last thing we'd want to fill in, because notice, six plus three plus one only adds up to 10. What about the other two? Well the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two, these are neither chocolate nor coconut. I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two way table. Two way table. On one axis, say the vertical axis, we could say, let me write this, so has chocolate, I'll write choc for short, and then I'll write no chocolate, choc for short. Then over here I could write coconut. I want to do that in white. I got new tools and sometimes the color changing isn't so easy. So this is coconut and then over here I'll write no coconut. Then let me make a little table. Make it clear what I'm doing here. So a line there and a line there and why not add a line over here as well. Then I can just fill in the different things. This cell, this square, this is going to represent the number that has coconut and chocolate. Well we are already looked into that, that's one, two, three, that's these three right over here. So that's those three right over there. This one right over here, it has chocolate, but it doesn't have coconut. Well that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. Then this box would be it has coconut, but no chocolate. Well how many is that? Well coconut no chocolate that's that one there. Then this one is going to be no coconut and no chocolate. We know what that's going to be. No coconut and no chocolate is going to be two. If we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total, I could write total, and if I total it vertically, so three plus one this is four, six plus two is eight. So this four is the total number that have coconut that has chocolate and doesn't have chocolate. That's three plus one. This eight is the total that does not have coconut. No coconut. So the total of no coconut, and that of course is going to be the six plus this two. We could total horizontally. Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three? This is the total amount no chocolate. That's this one plus two. Anyway, hopefully you found that interesting. This is just different ways of representing the same information.