Suppose I have a group of $N$ people, attending a series of $M$ events, and (for simplicity) let's assume the overall attendance happens to be the same at each event, say $A$ people (ranging between 1 and $N$), or as a fraction $a = A/N$ (ranging between 0 and 1).
However, it may not be the same people attending each event. What does knowing the attendance tell me about the individual attendance behaviour? In particular, can I draw any conclusions about the regularity of event attendance (i.e. the number of events an individual has been to)?
For example, with fractional attendance $a$, it may be that
(1) the same $a N$ people attended every event in the series ($M$ events), and the other $(1-a) N$ people never attended, giving an average of $a$.
(2) Or it may mean that all $N$ people attended $a M$ events, also giving an average of $a$.
My main question: Can you show that $a N$ people attended at least $a M$ events? More generally, given the average $a$ what is the minimum number of people $X$ (out of $N$), that attended at least $Y$ events (out of $M$)?
I would have thought that this is relatively straight forward, but I couldn't find or come up with an answer.
Extension 1: How would this change if the attendance at each event is variable? Would does the discrete distribution of attendance values $a_i=A_i/N$ ($i=1..M$), including the average $a = \sum_{i=1}^M a_i / M$, standard deviation, etc tell me about attendance pattern?
Extension 2: I would also have thought that there are some combinatorial statements like: "If I pick $a_1, a_2, a_3, ..., a_n$ out of $N$, the minimum overlap between $a_1, a_2, a_3, ..., a_n$ is such-and-such".