# Standard probability space and random variables.

Problem :

Let $$\Omega:=[0,1]$$

How to define two independent random variables, which describe coin toss on the above omega?

My idea:

$$\Omega=[0,1]$$

$$\Sigma = \mathbb{B}([0,1])$$

Consider the Lebesgue measure

1-reverse coin 2-obverse coin

$$X(w)= \begin{cases} 0 &\text{when } w \in [0,1/2) \\ 1 &\text{when } w\in [1/2,1) \end{cases}$$

$$Y(w)= \begin{cases} 1 &\text{when } w \in [0,1/4) \cup [1/2,3/4) \\ 0 &\text{when } w\in [1/4,1/2)\cup [3/4,1) \end{cases}$$

It seems, that $$P(X=1)=P(X=0)=P(Y=0)=P(Y=1)=1/2$$

And, for example $$P(X=0)P(Y=1)=1/4=P(X=0,Y=1)$$