Determining the measurable space and density function for an event

Question

I have 10 balls. 3 of the balls are faulty. I choose 5 balls without returning them back to the stack of balls.

$X$ $=$ "the amount of faulty balls"

• Is the measurable space of this scenario (10, 2) or something entirely different?

• How can I build a probability density function for $X$?

Answer 1

Let $A:=\{1,2,3,4,5,6,7,8,9,10\}$.

You can take $\Omega=\{\omega\in\wp(A)\mid |\omega|=5\}$ as sample space and $\wp(\Omega)$ as $\sigma$-algebra.

$\Omega$ has $\binom{10}5$ elements and the events $\{\omega\}\in\wp(\Omega)$ are equiprobable, so that $P(\{\omega\})=\binom{10}5^{-1}$ for each $\omega\in\Omega$.

This determines $P$ so now probability space $(\Omega,\wp(\Omega),P)$ is constructed.

Now let's say that the balls corresponding with the numbers $1,2$ and $3$ are faulty.

Then random variable $X:\Omega\to\mathbb R$ is prescribed by: $$\omega\mapsto|\omega\cap\{1,2,3\}|$$

Evidently $X$ takes values in $\{0,1,2,3\}$ and you can find the probability mass function (not density) of $X$ by finding the $p_X(k):=P(X=k)$ for $k=0,1,2,3$.