Problem Solving

Can anybody please explain the approach to solving problems in overlapping sets?

It'd be great if we could discuss a few examples.
I know of just one formula for such problems and I can solve such problems only when its straight forward

P(A u B u C) : P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C) + P(A n B n C)

1st example problem -

In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% 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?

Expert advice preferred

this is not the expert's post

Set is a group of elements. Overlapping sets can be named "overlapping" when there participate at least two independent sets. Two or more independent sets overlap when their elements become shared - the elements in overlapping sets will be equal or greater in the number of those in the independent sets; this way sets overlap. Namely, each overlapping set becomes a new set with the elements borrowed from the independent sets.

crimson2283 wrote:Can anybody please explain the approach to solving problems in overlapping sets?

It'd be great if we could discuss a few examples.
I know of just one formula for such problems and I can solve such problems only when its straight forward

P(A u B u C) : P(A) + P(B) + P(C) - P(A n B) - P(A n C) - P(B n C) + P(A n B n C)

given: the independent sets --> 50% of those asked liked product 1 - Set A, 30% liked product 2 - Set B, and 20% liked product 3 - Set C // an overlapping set --> 5% of the people in the survey liked all three of the products - Set D (borrowed from A, B and C), find percentage of the survey participants liked more than one of the three products -?

solution: note there's only 85% of survey participants who liked at least one of three products - at least means, they could like one product, two products, or three products too. If we subtract 5% of those howl liked all three products we should get --->
45% liked product 1 - Set A, 25% liked product 2 - Set B, and 15% liked product 3 - Set C// total makes 85% (this much participated in the survey BTW ) So 5% from each set liked all three products, and 5% from each set should like also two products. if we received 5%+5% for those who liked all three and two products we get exactly the percentage of participants who liked at least more than one product (two and three).

1st example problem -

In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% 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?

Expert advice preferred

crimson2283 wrote:In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% 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) 5

B) 10

C) 15

D) 20

E) 25

Here is the formula for 3 overlapping groups in which sometimes 2 of the groups overlap and sometimes all 3 groups overlap:

T = G1 + G2 + G3 - (those in 2 of the groups) - 2*(those in all 3 groups)

The trick with overlapping group problems is to subtract the overlap. When we add together everyone who likes product 1, everyone who likes product 2, and everyone who likes product 3, those who like exactly 2 of the products will be counted twice, so they need to subtracted from the total once. Those who like all 3 products will be counted 3 times, so they need to be subtracted from the total twice.

In the problem above:
T = 85
G1+G2+G3 = Product 1 + Product 2 + Product 3 = 50+30+20
Those who like exactly 2 of the products = x
Those who like all 3 products = 5

Plugging into the formula, we get:

85 = 50 + 30 + 20 - x - 2*5
x = 5 consumers who like exactly 2 of the products.

Since 5 consumers like exactly 2 of the products and 5 like all 3 products, the number who like at least 2 of the products = 5+5 = 10.

The correct answer is B.