I have to compute the expected value that a subset is bad, with the probability any set is bad being $\frac{1}{t^3}$. However, I have no idea how many sets there are, only the probability that any individual one is bad. So when I use my expected value equation, how do I account for this? Is it simply an infinite sum?
EDIT
The full problem is I have a set $V$ and a family of sets $F$ such that each set in $F$ contains a three element subset of $V$. The probability of any element in $V$ being 'bad' is $\frac{1}{t}$, and any set in $F$ is 'bad' if all three elements in that set are 'bad', hence $\frac{1}{t^3}$. I want to find the expected number of bad sets in $F$ but I am not given the size of $F$ or $V$.
This is a homework problem but it is for practice not a grade.
I think the answer is $|F|\cdot\dfrac {1}{t^3}$. Let $F=\{F_1,\dots,F_{|F|}\}$. The expected number of bad sets in $\{F_1\}$ is $\dfrac {1}{t^3}$, the expected number of bad sets in $\{F_2\}$ is $\dfrac {1}{t^3}$, and so on. By the linearity of expectation, the expected number of bad sets in $F$, which is the disjoint union of the $\{F_i\}$, is $|F|\cdot\dfrac {1}{t^3}$.