3 set venn diagram

**sfspitfire23** · September 10th 2010, 05:17 PM

Hi guys,

A group of 15-18 year old kids were studied. 68% have a television, 92% have a CD/tape player, and 32% have a computer. Only 4% have none of the above. 9% have a computer and a CD/tape player but no television 44% have a TV and a CD/tape player but no computer, and 1% have a TV and computer but no CD/tape player.

How would I fill out this venn diagram?

Work:

Let A= TV, B= CD/tape, C= computer.
(Let u= union and n=intersection)
So, we know that (AnB)= .44, (AnC)= .01, and (BnC)=.09.
We must find (AnBnC).

Formula:
Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps)+(all three overlap) + Neither.

So, 1=.68+.92+.32-(.54)+X+.04

We get (all three overlap)= -.42. This does not make sense...

I know these formulas but am unsuer how to apply them:
Exactly 2 sets=(sum of 2-group overlaps)-3(all three overlap)
More than 3 sets=(sum of 2-group overlaps)-2(all three overlap).

Am I way off here? What formulas can I use for this?
Thanks

**Soroban** · September 10th 2010, 07:33 PM

Hello, sfspitfire23!

This is a tricky one . . .

$\text{A group of youths was studied. 68\% have a television,}$ . . $\text{92\% have a CD/tape player, and 32\% have a personal computer.}$

$\text{Only 4\% have none of the above.}$
$\text{9\% have a PCr and a CD player but no TV.}$
$\text{44\% have a TV and a CD/ player but no PC.}$
$\text{1\% have a TV and PC but no CD player.}$

$\text{How would I fill out the Venn diagram?}$

Construct the Venn diagram and fill in the known quantities.

Code:

      o---------------------------------------------------o
      |                                                   |
      |     o-----------------------o                     |
      |     | TV                    |                     |
      |     |       b       o-------+---------------o     |
      |     |               |       |            CD |     |
      |     |               |  44   |       c       |     |
      |     |               |       |               |     |
      |     |       o-------+-------+-------o       |     |
      |     |       |       |       |       |       |     |
      |     |       |   1   |   a   |       |       |     |
      |     |       |       |       |       |       |     |
      |     o-------+-------+-------o       |       |     |
      |             |       |           9   |       |     |
      |             |       |               |       |     |
      |             |       o---------------+-------o . . |
      |             |     d                 |             |
      |             | PC                    |             |
      |   4         o-----------------------o             |
      |                                                   |
      o---------------------------------------------------o

Use $a,b,c,d$ in the unknown spaces.

In the TV-circle, the total is 68.
. . $a + b +45 \:=\:68 \quad\Rightarrow\quad a + b \:=\:23 \;\;[1]$

In the CD-circle, the total is 92.
. . $a + c + 53 \:=\:92 \quad\Rightarrow\qud a + c \:=\:39 \;\;[2]$

In the PC-circle, the total is 32.
. . $a + d + 10 \:=\:32 \quad\Rightarrow\quad a + d \:=\:22\;\;[3]$

In the three circles, the total is 96.
. . $a + b + c + d + 54 \:=\:96 \quad\Rightarrow\quad a + b + c + d \:=\:42 \;\;[4]$

Subtract [2]-[1]: . $c - b \:=\:16\;\;[5]$

Subtract [2]-[3]: . $c - d \:=\:17\;\;[6]$

Subtract [4]-[1]: . $c + d \:=\:19\;\;[7]$

Add [6]+[7]: . $2c \:=\:36 \quad\Rightarrow\quad \boxed{c \,=\,18}$

Substitute into [7]: . $18 + d \:=\:19 \quad\Rightarrow\quad \boxed{d \,=\,1}$

Substitute into [5]: . $18 - b \:=\:16 \quad\Rightarrow\quad \boxed{b \,=\,2}$

Substitute into [1]: . $a + 16 2 \:=\:23 \quad\Rightarrow\quadn \boxed{a \,=\,21}$

Now you can fill in the rest of the Venn diagram . . .

**sfspitfire23** · September 11th 2010, 07:28 PM

Ah! Thanks! I see now. thanks for the detailed explanation.

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