I need to calculate the proportion of people that are exposed to an advertisement on either tv, radio, or online, based on a subset of proportions. I have the following data:
- the proportion of people exposed to a tv ad
- the proportion of people exposed to a radio ad
- the proportion of people exposed to an online ad
- the proportion of people exposed to a tv ad AND an online ad
- the proportion of people exposed to a tv ad AND a radio ad
- the proportion of people exposed to an online ad AND a radio ad
In other words, I'm trying to find the proportion of people exposed to any type of ad, based only on the proportion of people exposed to each type of ad, and the proportion of people exposed to each pairwise combination of ad types.
After a bit of googling this seems like a job for set theory. I found this equation:
$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|.$$
Based on this equation, in order to get $|A \cup B \cup C|$ (the proportion of people seeing any of the three ad types) it seems like you need to know $|A \cap B \cap C|$ (proportion of people seeing to all three types of ad) in addition to $|A|$, $|B|$, and $|C|$ (proportion of people seeing each ad) and $|A \cap B|$, $|A \cap C|$ and $|B \cap C|$ (proportion of people seeing all pairwise combinations of ads).
But this doesn't make sense to me - if I draw a Venn diagram of two circles based on the values of $|A|$, $|B|$, and $|A \cup B|$, then it seems that there's only one way to draw a third circle that satisfies the values of $|C|$, $|A \cap B|$ and $|B \cap C|$. In other words, the values of $|A \cap B \cap C|$ and $|A \cup B \cup C|$ should depend entirely on the values of $|A|$, $|B|$, $|C|$, $|A \cap B|$, $|A \cap C|$ and $|B \cap C|$, and I should be able to calculate $|A \cup B \cap C|$ using only the data I have available.
My question: is my logic correct here? And if so, what equation should I use to calculate $|A \cup B \cup C|$ using the data I have available? Also, is there a more general equation or algorithm I can use to calculate the union of $>3$ sets, given only the proportions for each set separately and the proportions for all pairwise intersections?
Suppose you know that $$|A|=|B|=|C|=2\ \text{ and }\ |A\cap B|=|A\cap C|=|B\cap C|=1;$$ from that, how big do you think $|A\cup B\cup C|$ is?
Now check your answer against these two examples:
$$\text{Example 1.}\ \ A=\{1,2\},\ B=\{1,3\},\ C=\{2,3\}$$ $$\text{Example 2.}\ \ A=\{1,4\},\ B=\{2,4\},\ C=\{3,4\}$$ In Example 1, $|A\cap B\cap C|=0;$ in Example 2, $|A\cap B\cap C|=1.$