Suppose there are $n$ couples in a party. What is the way of choosing a man and a woman who are not a couple.
I can choose a woman in $n$ ways ($E_1$), and I am left with $n-1$ choices for a man not her husband ($E_2$). Now, I cannot decide if I should use the sum rule or the product rule, i.e are the total possibilities $2n-1$ or $n^2-n$. $E_1$ and $E_2$ do not seem to be independent, as the event E_1 automatically determines the set of $E_2$. The sum rule sounds possible, as $E_1$ and $E_2$ cannot occur simultaneously (because I will have to choose a woman first and then choose a man from the remaining set).
I am having trouble when to use sum and product rules in general.
Best Answer
The sum rule would tell you how many ways to pick either a woman, or a man who is not her husband; that doesn't really make sense at all (how can you only pick "a man who is not her husband"?), so the sum rule cannot be applicable here.
You can also argue by overcounting and then compensating, as follows: there are $n^2$ ways to pick a man and a woman ($n$ ways to pick the woman, $n$ ways to pick the man; you want to pick both, so you multiply them). But of these $n^2$ ways to pick a man and a woman, $n$ of them pair up a woman with her husband. So $n$ of the $n^2$ selection are "no good", so we must take them off. This gives a total of $n^2-n$ possible ways of picking the couple.