In a survey of political preference, $78$% of those asked were in favor of at-least one of the proposals: I, II and III. $50$% of those asked favored proposal I, $30$% favored proposal II, and $20$% favored proposal III. If $5$% favored all the three proposals, what % of those asked favored more than one of the three proposals?
A homework problem involving counting using a special case of inclusion/exclusion
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1Could you please use an even _less_ useful title for your question? – 2010-10-29
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0I have the opposite reaction -- :-) -- to that of Tobias Kienzler: please choose a better title for your question. This one could apply to any question, so is absolutely useless. – 2010-11-28
3 Answers
The simplest way is probably to draw Venn diagrams and stuff, but here's how you do it using the inclusion-exclusion principle.
For any three sets $A,B,C$, we have $$|A \cup B \cup C| = S_1 - S_2 + S_3,$$
where $S_1 = |A|+|B|+|C|$, $S_2 = |A \cap B| + |B \cap C| + |C \cap A|$, and $S_3 = |A \cap B \cap C|$.
Letting $A,B,C$ denote the obvious sets, you are given the values of $|A \cup B \cup C|$, $S_1$ and $S_3$ so you can solve for $S_2$.
Now you are asked to find $x = |(A \cap B) \cup (B \cap C) \cup (C \cap A)|$, which we can expand with the same formula (using the fact that $(A \cap B) \cap (B \cap C) = A \cap B \cap C$ etc.) as $$x = |A \cap B| + |B \cap C| + |C \cap A| - 3|A \cap B \cap C| + 3|A \cap B \cap C| = S_2 - 2S_3.$$
If you do this right, and assuming I've done it right, the answer should be 17%.
hmm, I got $17$, not $27$...
If we count $|A|+|B|+|C|$, then we overcount intersections of any $2$ sets twice, and intersections of all three $3$ times. Then, if we subtract $|A \cup B \cup C|$, we get the intersections of any $2$ sets once and $3$ sets $2$ times. (most easily seen by drawing a Venn diagram). Subtracting the $3$-set intersection once more gives $100-78-5=17$.
We have (using $ab$ for those who favor I and II, $n$ for those who favor none, etc)
$a+b+c+2ab+2ac+2bc+3abc=?$
$a+b+c+2ab+2ab+2ac=?$
$a+b+c+ab+ac+bc+abc=?$
$a+b+c+ab+ac+bc=?$
and so?