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In a consumer survey, 85 percent of those surveyed liked at [#permalink]

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24 Nov 2005, 17:24

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In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?
_________________

Whether you think you can or think you can't. You're right! - Henry Ford (1863 - 1947)

In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

A=0.5
B=0.3
C=0.2
AUBUC=.85
AandBandC=.05

AUBUC=A+B+C-(people who liked at least two) - AandBandC
.85=1-X-.05
x=1-.85-.05=.1

In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

A=0.5 B=0.3 C=0.2 AUBUC=.85 AandBandC=.05

AUBUC=A+B+C-(people who liked at least two) - AandBandC .85=1-X-.05 x=1-.85-.05=.1

Please correct me if I am wrong, but as per what I have understood out of sets,
n(A U B U C) = n(A) + n(B) + n(C) - n(A U' B) - n(B U' C) - n(A U' C) + n(A U' B U' C)

ie. it should be

AUBUC=A+B+C-(people who liked at least two) "+" AandBandC

You get different results when you count people together or when you count them seperately. For example, say there are three items A, B, and C. If 3 people like A and B, 3 people like B and C, and 3 people like A and C. The number of people who like at least two items are not necessarily 9. Perhaps there are only 7 people who like at least two items, because 1 person likes all the three of them. (2 A and B only, 2 B and C only, 2 A and C only, and 1 like all of them.) This is why the two formulas are different. When you count them seperately and add them up, you have twice overcounted the center when they intersept each other.

In other words, the original formula is this:
N(AUBUC) = N(A) + N(B) + N(C) - N(A n B) - N(A n C) - N(C n B) + N(A n B n C)
When you subsititute N(A n B) + N(A n C) + N(C n B) = N(at least two) + 2N(A n B n C)
You will get
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - 2N(A n B n C) + N(A n B n C)
Which means
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - N(A n B n C)
_________________

Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.

You get different results when you count people together or when you count them seperately. For example, say there are three items A, B, and C. If 3 people like A and B, 3 people like B and C, and 3 people like A and C. The number of people who like at least two items are not necessarily 9. Perhaps there are only 7 people who like at least two items, because 1 person likes all the three of them. (2 A and B only, 2 B and C only, 2 A and C only, and 1 like all of them.) This is why the two formulas are different. When you count them seperately and add them up, you have twice overcounted the center when they intersept each other.

In other words, the original formula is this: N(AUBUC) = N(A) + N(B) + N(C) - N(A n B) - N(A n C) - N(C n B) + N(A n B n C) When you subsititute N(A n B) + N(A n C) + N(C n B) = N(at least two) + 2N(A n B n C) You will get N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - 2N(A n B n C) + N(A n B n C) Which means N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - N(A n B n C)

Shouldn't do you be adding P1andP2andP3 to 10% giving an answer of 15%?

that 10% is resulted with the inclusion of that percentage figure for (P1 and P2 and P3).

the required % = % liked 2 products + % liked 3 or all products
= % of people liked products (1 & 2 +2 & 3+ 1 & 3) + % of people liked all 3 (P1 and P2 and P3)
= 5%+5%
= 10%

note: % of people liked products (1 & 2 +2 & 3+ 1 & 3) is calculated as under:

total % of people surveyed = % of people liked product 1 + % of people liked product 2 + % of people liked product 3 - % of people liked products (1 & 2) - % of people liked products (2 & 3) - % of people liked products (1 & 3) - 2 [% of people liked products (1, 2, & 3)]
85%= 50+30+20 - p(1 & 2) - p(1 & 2) - p(2 & 3) - p(1 & 3) - 2 (5%)
p(1 & 2) + p(1 & 2) + p(2 & 3) + p(1 & 3) = 100-85-10 = 5%