Q:

Finding Density and Distribution of a random variable

I have the next problem:
An experiment is consisted by dropping two red balls simultaneously into 4 numbered boxes in the way that the balls must to fall in at least one box, and they have the same probability to fall in any box. Let $X$ the random variable which denotes the number of balls that falled into the Box #1. Find the density and distribution function for $X$.
The matter here is that I'm trutly lost about this one, because I don't even know which one is my probabilistic space $\Omega$, to define de probability of the event "The Ball $i$ falled into the $j$ box. It's obvious that this Random variable $X$ is discrete.

A:

$\Omega$ is the set of all possible outcomes of your experiment.
So you can for instance write it as $\{1,2,3,4\}^2$ (where the numbers $1$ to $4$ designate the boxe into which the ball falls). Hence, a possible outcome of the experiment is for instance $(1,1)$ (both balls in box 1) or $(1,3)$ (ball 1 in box 1, ball 2 in box 3), etc...
Next question is the probability you want to use. As long as you keep track of the number of each ball, it is easily seen from your assumption ``the balls must to fall in at least one box, and they have the same probability to fall in any box'', that all configurations have the same probability to happen, so that for any $i,j$ in $\Omega$, $\mathbb{P}(i,j) = \frac{1}{16}$ (since $|\Omega| = 4^2 = 16$). 
It is easy then to see that your random variable $X$ takes values in $\{0, 1,2\}$, and that, for instance $\mathbb{P}(X = 2) = \mathbb{P}((1,1)) = \frac{1}{16}$.
Computing the other two probabilities for $X$ is just a matter of counting all the configurations well. From then you will easily find the distribution function and the density you are looking for.

