Short question: Is independence of random indicator variables a necessary assumption to derive the Binomial distribution ?
But far as I can see, using the Definition (that is at the same time a theorem) from below, taken from Snells probability book, page 144, we can do without this assumption and prove that the random indicator variables are independent! (as opposed to, for example, the geometric distribution, were we need independence to derive it).
EDIT: A better explanation: Do we in the text below assume the variables are independent, or to we deduce it from their definition ?
To me it seems the latter is the case, since we concretely define the $X_j$ on a given sample space, so we can use the definition of independence of variables to test if they are indeed independent or not - so independence is proven, not assumed.