On the random variables and conditional probability

Question

Let $X$ and $N$ be random variables. Suppose that $N$ is binomial $Bin(M, q)$ and, given $N$, $X$ is $Bin(N,p)$. How to show that $X$ is $Bin(M,pq)$ ?

Answer 1

The proof is given on the Binomial distribution Wikipedia page

Answer 2

We can get the result by a simple pictorial-combinatorial interpretation.

We have $M$ cells. Alice paints randomly each cell with probability $q$. Bob places randomly a ball in each cell with probability $p$ (all events are independent).

Let $N$ be the amount of painted cells, and $X$ the amount of painted-and-full cells.

Clearly, $N \equiv Bin(M,q)$ . Also $X \equiv Bin(M,pq)$.

Finally, notice that, when given $N$ (painted cells) to get $X$ is equivalent to count how many balls (with probability $p$) are placed in $N$ cells. That is, $X|N \equiv Bin(N,p)$.