# Notation of Probability space

Assume we have red and a black cube (normal cubes with 6 sides). We roll these two dice. Is it true that the sample space is $$\Omega = \left\{ (r,s) \mid r \in \{ 1, \dots, 6\}, s \in \{ 1, \dots, 6\} \right\}?$$

I'm now interested in the event $$A =$$'The red cube shows an even number' for example. Can I define the set $$A$$ like
$$A = \{(r,s) | r \in \{2,4,6\}, s \in \{ 1 \dots 6\} \}?$$ How can I define $$A$$'s probability density function and probability measure for this probability space?

your event $$A$$ is defined correctly. Before defining a probability measure you have to define a simga-field. intuitively, the sigma-field contains all sets which you want to be able to assign a probability to. Most often we simply choose the power set of $$\Omega$$ as the corresponding sigma field $$\mathcal F$$, i.e. $$\mathcal F = \mathcal P (\Omega).$$
By doing so we are able to define a probability measure $$\Bbb P$$ on $$\mathcal{F}$$ which assigns a probability to every possible event. Since every event has the same probability and there is a total of 36 events we could define: $$\Bbb P: \mathcal F \rightarrow [0,1]: A \mapsto \frac{|A|}{36}$$ In particular, if we want to compute the probability that the red dice is even we have to count all events where this is the case. It is easy to verify that in 18 ($$= 3*6$$) cases the red dice is red. Therefore $$\Bbb(P)(A) = \frac{18}{36} = \frac 12,$$ in this case.
• I think in this case, $F=\{\phi ,\Omega\}$ and can only calculate $P(\phi )=0$ $P(\Omega)=1)$. an example: let we have one cube, that $D=\{ 1,2\})$ and $B=\{ 3,4,5,6\}$ have same chance. what is the $P(A=\{2,4,6\})$? (note by that info i told $F=\sigma \{D,D^{c} \}$) we can not talk about the events like $A$ that have no information about it. since we can not calculate it – masoud Mar 26 '19 at 10:25