Question 1203550: In a recent survey, a statistician reported the following data.
13 persons liked brand A.
20 persons liked brand B.
13 persons liked brand C.
9 persons liked brands A and B.
4 persons liked brands A and C.
8 persons liked brands B and C.
2 persons liked all three brands.
3 persons liked none of the three brands.
A truthful statistician was asked to find out how many people were interviewed. What was this statistician's answer?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
In a recent survey, a statistician reported the following data.
13 persons liked brand A.
20 persons liked brand B.
13 persons liked brand C.
9 persons liked brands A and B.
4 persons liked brands A and C.
8 persons liked brands B and C.
2 persons liked all three brands.
3 persons liked none of the three brands.
A truthful statistician was asked to find out how many people were interviewed. What was this statistician's answer?
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For all such and similar problems, there is a common/general method of their solutions.
This method is called "Inclusion-Exclusion Principle".
It says that if there is given info about the sizes of 3 finite subsets A, B and C of a universal set U,
and there is given info about the sizes of their in-pairs intersections AB, AC and BC,
and there is given info about the size of their triple intersection ABC,
then the number of elements in the union (A U B U C) is the alternate sum
n(A U B U C) = n(A) + n(B) + n(C) - n(AB) - n(AC) - n(BC) + n(ABC).
In your case it gives for the union
n(A U B U C) = 13 + 20 + 13 - 9 - 4 - 8 + 2 = 27.
To get the ANSWER to the problem's question,
you need add 3 persons that are outside of the union (A U B U C)
27 + 3 = 30.
ANSWER. 30 people were interviewed.
Solved.
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As a reference to the Inclusion-Exclusion principle, see this Wikipedia article
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
for tutor @ikleyn....
You are missing one close parenthesis in your statement of the general principle that might cause confusion to beginning students....
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