How likely is it that this distribution is random? This is not a homework or similar, this is the real world, I am preparing some stats for a meeting I will attend.
I am a member of an association that has 26 members. There are two types of members, A (14 out of 26) and B (12).
The board consists of 5-6 members elected for one year at the time. The board members are split into two categories, "regular" board members and deputies. During the last 10 years, there have been 56 "man-year" on the board (that is, the board has had on average 5,6 members per year).
Out of the 56 man-years, 44 have been regular members and 12 for the deputies.
Out of the regular members' 44 man-years, 40 have been of member type A while 4 member type B.
OTOH just 1 out of the 12 man yeardeputies was A, while 11 was B.
Membership of the board should be randomly distributed. Everyone has one vote and everyone is eligible.
I would like to know how (un)likely it is that this distribution is random, for all three cases. How do I calculate that? 20 years since I studied econometrics and probability and I don't know where to start (nor finish...)
 A: If you really think that membership of the board should be randomly distributed (see Bjorn's comment -- I agree with him), and that each personXyear is iid from a distribution (importantly, you're assuming that me last year is a completely different person as me this year, and so on), and you are interested in tests of the form A/B vs ordinary/deputy, or A/B vs on board/not on board, or A/B vs ordinary/not ordinary, etc, then I'd probably just set up a bunch of 2x2 contingency tables. Of course, there are many caveats here, but if this is for a meeting and to make some points clear in that meeting, I think this approach is fine (if you were seeking to publish a peer-reviewed paper on lack of diversity of A and B people in the workplace, then certainly this is not good enough). So for example, let's say I want to test A/B vs regular/not regular. Then my table, using your numbers, is as follows:
\begin{array}{|c|c|c|c|}
\hline
& \text{reg} & \text{not reg} & \text{total} \\ \hline
 \text{A}  & 40 & 100 & 140\\ \hline
 \text{B}& 4 & 116 & 120 \\ \hline
\text{total} & 44 & 216 & 260 \\ \hline
\end{array}
Then, your null hypothesis is that being in A or B has no effect on the probability of being reg vs not reg, so intuitively, you'd expect $44/216 \approx .2$ of both groups to be reg (i.e. $140*.2 = 28$ group A individuals and $120*.2 = 24$ group B individuals to be reg), but clearly, we don't observe that. The way to formalize that is to do a chi squared test. Check out the wikipedia link on chi squared tests to see how to implement them, and all statistical programming languages should easily allow you to perform these. You can also find online tools to quickly perform this analysis.
