Overlapping Sets Chapt 7 pg 95

[kimd6746]

I'm stone cold stumped on the the very first example on pg 95

"30 people are in a room. 20 of them play golf. 15 of them play golf and tennis. If everyone plays at least one of the two sports, how many of the people play tennis only?"

From this I surmise the following categories.

1. My sum is 30
2. People who play both sports is 15
3. Conversely, people who play only one sport is 15
4. People who play golf is 20. (does not say only play golf, so this must be where the overlap exists right?)
5. Conversely, doesn't that mean the people who don't play golf (eg play tennis) is 10? (Another overlap?)

According to the book, the intersection of A to B is 8. That box to me represents people who play both A and B. Since the book doesn't specifically label A and B, I'm assuming A = Golf and B = Tennis. How did they get to 8? Shouldn't A and B = 15??? They also say the total of B is 22. How was this calculated? I am utterly lost on this example's logic. Can someone please explain how to setup this doubleset matrix and what the correct categories should be?

[StaceyKoprince]

Read just a little more thoroughly and you won't be stumped. :)

The first problem, on the top half of the page, is not the same problem as the second one (on the bottom half of the page).

In the first, you've got 20 people in the golf category and 15 in the both category. Down below, notice you're now talking about integers, not people, and you've got 15 in set A, 22 in set B, and 8 in the both category - that's a different problem!

[kimd6746]

OK, I thought I was losing it. I understand the bottom half of pg 95 but I'm still stuck on the first problem. How would you solve it?

My thought process was as follows:

1) Total people is 30.
2) Total people who play golf is 20.
3) People who play both tennis and golf is 15.
4) that makes people who only play golf 5.
5) But I can't derive the missing intersections as per below.

GOLF TENNIS TOTAL
ONLY 5 ? ?
BOTH 15 ? ?
20 10 30

I'm sure this overlapping set matrix can't be right but this is the most logical setup I could think of. How should this have been setup correctly?

[kimd6746]

I think I answered my own question.

I think what I was missing was that I was ignoring the statement of "if everyone plays at least one of the two sports." If I had factored this into my matrix, I would realize that the intersection of No Tennis and No Golf must equal zero.

1) Total people is 30
2) People who play golf is 20
3) People who don't play golf must be 10
4) People who play both is 15
5) That means people who play golf but not tennis is 5
6) Since everyone must play one of the two sports, intersection of no tennis and no golf must be 0.
7) Therefore, people who play tennis but don't play golf must be 10.


..............GOLF....NO G.....TOTAL
TENNIS.....15........10.........25
NO T..........5..........0...........5
TOTAL.......20........10.........30

Did I get this right??? :idea: :idea: :idea:

[StaceyKoprince]

You've got it! And that kind of subtle wording on the test is very common - they won't tell you that some category is zero, but they'll tell you that everybody plays at least one sport; you've got to make the inference. Nice job!

[metman82]

Using a Venn diagram here is much easier I think.


It can be seen that it yields to 10.

I think the Venn Matrix can be used in easier question types or in questions with 3 Sets. Am I right?

[jnelson0612]

Yes, the Venn is the best way to go when you have three overlapping groups. For example, students can take Spanish, French, or German. Some take one language, some take two, and some take all three.

You *can* use a Venn on a fairly simple problem with two overlapping sets; however, I'm a bigger fan of the double-set matrix. I think it's easier to use and organizes the information better. Sometimes, too, you can just use a simple equation on an easy overlapping sets problem:

Total=Group 1 + Group 2 - both + neither