Not quite.kulsim wrote:Hi! Apologies if this sounds trivial but just trying to make sure I am getting this fully. I now understood why we got 25% and just trying to think about this problem in reverse. If we were asked to maximize the ABC, would the answer be simply 75%?
To make the reasoning easier to see, let's assume that everyone in the survey likes at least one of the 3 fruits.
In this case, we can use the following formula:
T = A + B + C - (AB + AC + BC) - 2(ABC)
The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.
In the problem above:
T = 100%.
A = 70%.
B = 75%.
C = 80%.
Plugging these values into the formula, we get:
100 = 70 + 75 + 80 - (AB + AC + BC) - 2(ABC).
100 = 225 - (AB + AC + BC) - 2(ABC)
(AB + AC + BC) + 2(ABC) = 125.
MINIMUM value of ABC:
To MINIMIZE the value of ABC, we must MAXIMIZE the value of (AB + AC + BC).
Here, the maximum value of (AB + AC + BC) = 75, as shown in my post above:
75 + 2(ABC) = 125
2(ABC) = 50
ABC = 25%.
Thus, the least possible value of ABC = 25%.
MAXIMUM value of ABC:
To MAXIMIZE the value of ABC, we must MINIMIZE the value of (AB + AC + BC).
If AB + AC + BC = 0, we get:
0 + 2(ABC) = 125
2(ABC) = 125
ABC = 62.5%.
Thus, the greatest possible value of ABC = 62.5%.
If it's possible that some people like NONE of the 3 fruits, we can use a variation of the formula above:
T = A + B + C - (AB + AC + BC) - 2(ABC) + None.
In this case -- since the percentage who like all 3 fruits cannot exceed the percentage who like cherries -- the greatest possible value of ABC is 70% (the percentage attributed to C).
The least possible for (AB + AC + BC) is still 0.
Plugging all of the percentages into the amended formula, we get:
100 = 70 + 75 + 80 - 0 - 2(70) + None
100 = 85 + None
None = 15%.
Implication:
If 15% like none of the 3 fruits, then it's possible that 70% like all 3 fruits.
For more practice with triple-overlapping groups, check my post here:
https://www.beatthegmat.com/og-13-178-vi ... 11188.html