SOLUTION: JOBITA Nigeria limited produces three brands P,Q and R of coffees. in order to determine the marketability of the product, some samples were sent out. on analysis of the customers

Question 1113348: JOBITA Nigeria limited produces three brands P,Q and R of coffees. in order to determine the marketability of the product, some samples were sent out. on analysis of the customers opinions, the following information were obtained. 250 customers liked brand P, 250 like brand Q and 283 like brand R. 50 customers like brands P and Q, 65 customers liked brands P and R, 55 liked Q and R. if 625 customers were interviewed, find the number of customers who like all the
(i)three brands
(ii) exactly on brand
(iii) at least two brands
(iv) brands P and Q but not R
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Nigeria limited produces three brands P,Q and R of coffees.
in order to determine the marketability of the product, some samples were sent out.
on analysis of the customers opinions, the following information were obtained.
250 customers liked brand P,
250 like brand Q and
283 like brand R.
50 customers like brands P and Q,
65 customers liked brands P and R,
55 liked Q and R.
if 625 customers were interviewed, find the number of customers who like all the:
:
No one wants this one, here is my take on it. No guarantee
Find the no. that only like one brand
250 - 50 - 65 = 135 liked P only
250 - 50 - 55 = 145 liked Q only
283 - 65 - 55 = 163 liked R only
---------------------------------
Total like 1 brand 443
therefore
625 - 443 = 182 like two or more brands
:
50 + 65 + 55 = 170 liked two brands
625 - 170 - 443 = 12 like all three brands
:
(i)three brands: 12
(ii) exactly one brand: 443
(iii) at least two brands: 182
(iv) brands P and Q but not R
50
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

I was curious to see what methods of solution other tutors might come up with for this problem....

There is a flaw in the analysis by the other tutor, leading to incorrect answers.

The easiest way to solve this problem is by using the inclusion-exclusion principle. That principle, with 3 brands, says that the total number of people sampled is equal to
(sum of numbers who liked one brand) minus (sum of numbers who liked two brands) plus (number who liked all three brands).

Note that in order to solve the problem we have to make the assumption (not stated in the problem) that every one of the people liked at least one of the brands.

So if x is the number of people who liked all three brands, then

So 12 people liked all three brands. Then, using that number with the given information, we can determine how many people liked which brands:

PQR: 12
PQ: 50-12 = 38
PR: 65-12 = 53
QR: 55-12 = 43
P: 250-38-53-12 = 147
Q: 250-38-43-12 = 157
R: 283-53-43-12 = 175

And now we can answer the specific questions that were asked.

(i) all three brands: 12
(ii) exactly one brand: 147+157+175 = 479
(iii) at least two brands: 12+38+53+43 = 146
(iv) P and Q but not R: 38

---------------------------------------------------------------

The inclusion-exclusion principle gets you to the answer quickly. But you can also find the number of people who like all three brands -- and thus finish the problem -- by using a Venn diagram with three circles representing brands P, Q, and R.

If x is again the number of people who like all three brands, then

PQR = x

PQ = 50-x
PR = 65-x
QR = 55-x

P = 250-(50-x)-(65-x)-x = 135+x
Q = 250-(50-x)-(55-x)-x = 145+x
R = 283-(65-x)-(55-x)-x = 163+x

The sum of all those numbers has to be the total number of people:

And from there you finish the problem as earlier.
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