Question

A special-interest group has conducted a survey concerning a ban on hand guns. Note: A rifle is a gun, but it is not a hand gun. The survey yielded the following results for the 1000 households that responded.

267 own a hand gun.
437 own a rifle.
493 supported the ban on hand guns.
136 own both a hand gun and a rifle.
206 own a rifle but no hand gun and do not support the ban on hand guns.
70 own a hand gun and support the ban on hand guns.
48 own both a hand gun and a rifle and also support the ban on hand guns.

How many of the surveyed households:

(a) only own a hand gun and do not support the ban on hand guns?


(b) do not own a gun and support the ban on hand guns?


(c) do not own a gun and do not support the ban on hand guns?

Answer 1

I like guns but im christian

Answer 2

a)109 of the surveyed households only own a hand gun and do not support the ban on hand guns.

b)328 of the surveyed households do not own a gun and support the ban on hand guns

c )112 of the surveyed households do not own a gun and do not support the ban on handguns.

Explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents those who own a hand gun.

-The set B represents those who own a rifle

-The set C represents those who support the ban of handguns.

-D represents those who do not own a gun and do not support the ban on hand guns.

We have that:

`A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)`

In which
`a` are those that only own a hand gun,
`A \cap B` are those who own both a handgun and a rifle,
`A \cap C` are those who own a hand gun and support the ban on handguns, and
`A \cap B \cap C` are those who own both a hand gun and a rifle, and also support the ban on hand guns.

By the same logic, we have:

`B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)`

In which
`b` are those who only own a rifle, and
`B \cap C` are those who own a rifle and support the ban on handguns.

`C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)`

In which
`c` are those who support the ban on hand guns and do not own any handguns.

This diagram has the following subsets:

`a,b,c,D(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)`

There were 1000 households surveyed, so:

`a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000`

We start finding the values from the intersection of three sets.

Solution:

48 own both a hand gun and a rifle and also support the ban on hand guns.

`A \cap B \cap C = 48`

70 own a hand gun and support the ban on hand guns:

`(A \cap C) + (A \cap B \cap C) = 70`

`A \cap C = 70 - 48`

`A \cap C = 22`

206 own a rifle but no hand gun and do not support the ban on hand guns.

`b = 206`.

136 own both a hand gun and a rifle:

`(A \cap B) + (A \cap B \cap C) = 136`

`A \cap B = 136 - 48`

`A \cap B = 88`

493 supported the ban on hand guns.

`C = 493`

437 own a rifle:

`B = 437`

267 own a hand gun:

`A = 267`

a) How many of the surveyed households only own a hand gun and do not support the ban on hand guns?

This is the value of
`a`, that we can find from the following equation:

`A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)`

`267 = a + 88 + 22 + 48`

`a = 267 - 158`

`a = 109`

109 of the surveyed households only own a hand gun and do not support the ban on hand guns.

b) How many of the surveyed households do not own a gun and support the ban on hand guns?

This is the value of
`c`, that we can find from the following equation:

`C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)`

`493 = c + 22 + (B \cap C) + 48`

Here, we also have to find
`B \cap C`, that are those who own a rifle and support the ban on hand guns. We can find this from the following equation:

`B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)`

`437 = 206 + (B \cap C) + 88 + 48`

`B \cap C = 437 - 342`

`B \cap C = 95`

So

`493 = c + 22 + (B \cap C) + 48`

`493 = c + 22 + 95 + 48`

`c = 493 - 165`

`c = 328`

328 of the surveyed households do not own a gun and support the ban on hand guns.

a) How many of the surveyed households do not own a gun and do not support the ban on hand guns?

This is the value of d, that can be found from the following equation:

`a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1000`

`109 + 206 + 328 + D + 88 + 22 + 95 + 40 = 1000`

`D = 1000 - 888`

`D = 112`

112 of the surveyed households do not own a gun and do not support the ban on handguns.