To show an application of the probabilistic method I want to colour the $n$ vertices of an undirected and connected graph with two colours, for example with red and green. I want to calculate the probability that a certain vertex has a certain colour.$\\$
For this I would like to show the following:
Colouring the vertices can be modeled by the probability space $(\Omega,\mathcal{A}, P)$ with $\Omega=\{red, green\}^{n}$, $\mathcal{A}=2^\Omega$ and $P=\mathcal{U}(\Omega)$, where $P(i-th\, vertex\, is\, red)=\frac{\frac{1}{2}2^{{n}}}{2^{{n}}}=\frac{1}{2}.\\$ This is the same when we flip an independet iid fair coin for every vertex in the graph. So the probability that $\tilde{P}(i-th\, vertex\, is\, red)=\frac{1}{2}$.
So, we know that $P(i-th\, vertex\, is\, red)=\frac{1}{2}=\tilde{P}(i-th\, vertex\, is\, red)$.
And now I have to prove the independece of vertices under $P$. How can I prove that?
HINT
One way is from the definition of probabilities of independent events. Can you show that if $R_i$ is the event that the vertex $i$ is red, then $$ \mathbb{P}[R_1 \cap R_2] = \mathbb{P}[R_1] \cdot \mathbb{P}[R_2] $$