Previously I made this post regarding an experiment where people are paired off into groups. For clarity I'll restate the problem here:
Imagine walking into a room with five teenage boys and five adult men who are their fathers. Each child has a unique parent and each father has a unique child. Your job is to guess which men are fathers of which boys based on family resemblance.
Now, imagine that a guesser carries out this experiment once and gets 20% of the matches correct. They carry out the experiment again (on ten new people) and get 80% of the matches correct. A third round with yet again new people results in 60% success. Through each experiment involving ten new people, let's call the number of correct guesses actually made by the guesser $\beta$.
My question is: How do we go about showing that the person doing the guessing is actually seeing the family resemblances and making correct guesses that are better than would be expected by chance? Obviously, a monte carlo simulation would help in this, but is there some statistical test already out there that covers this situation?
This is a different question from my previous post here.
Two points of confusion:
1.) First, this question has been through quite a few revisions, and many of the comments below might be responses to old versions of this question.
2.) For now I'd like to just stick to the $N = 10$ population. At some future point I might expand this to larger $N$, the possibility of siblings present, men and women who might mutually be parents and so forth.