probability of k dice out of N has number x

Question

Consider some unknown dice. By unknown I mean that it is not a regular dice and we do not know its properties. Roll $N$ dice at the same time. We know that the probability of $k$ dice all rolls number $x$ is $p$. In other words we know that if we roll $k$ dice instead of $N$ all $k$ of them will have number $x$ with probability $p$. What is the probability that if we consider every selection k from n dices ${N\choose k}$ we will have at least one selection all rolling number $x$? It might be a very easy question but I just cant get my head around this. Due to the overlap these selections will not be independent form each other and I kind of feel that this is a problem.

Answer 1

We can assume $N>k$ and $p>0$ because otherwise the problem is simple. If any group of $k$ dice have the same probability $p$ of all rolling $x$, we can conclude that each die has the same probability of rolling $x$ - for any dice $d_1$ and $d_2$ we pick some $k$ other dice, which have some probability $q>0$ of all being $x$, then since those dice plus $d_1$ have probability $p$, $d_1$ has probability $q/p$ and the same for $d_2$.

Therefore every die has probability $\sqrt[k]p$ of rolling $x$, and so the probability of some set of $k$ all rolling $x$ is the probability we have at least $k$ out of $N$ rolling $x$, which is the probability that a Binomial $(N,\sqrt[k]p)$ distribution is at least $k$, i.e. $$\sum_{j=k}^N\binom Nj(\sqrt[k]p)^j(1-\sqrt[k]p)^{N-j}.$$