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A fair color dice has two green sides, three red sides, and one blue side. It is rolled an infinite number of times. Let $$ S_n:=\text { number of } \text { ,,green''} - \text{ number of ,,blue'' } $$ be the difference of the number of dice result ,,green'' and the number of dice result ,,blue'', each up to the $n$th roll, $n \in \mathbb{N}$.
I want to find a $c \in \mathbb{R}$ such that $$ P\left(\frac{S_n}{n} \stackrel{n\rightarrow\infty}{\longrightarrow} c\right)=1. $$

My idea goes like this: We choose for $c$ the expected value from the random variable $S_1$: difference after one execution, because we use the strong law of large numbers. It is therefore valid $$ \mathbb{E}[S_1]=\mathbb{E}[X-Y]=\mathbb{E}[X]-\mathbb{E}[Y]=\frac{2}{6}-\frac{1}{6}=\frac{1}{6}=c $$ where $\mathbb{E}(X)$ is expectation of green and $\mathbb{E}(Y)$ is expectation of blue. Does this approach make sense? Do I have to verify $\mathbb{E}[S_1^2]<\infty?$

calculatormathematical
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    Yes that is correct. Here I assume you mean to define the Bernoulli random variables $X_i = 1_{{\mbox{roll $i$ yields green}}}$ and similar for $Y_i$ for $i \in {1, 2, 3, ...}$, and $Z_i=X_i-Y_i$. It is obvious that $E[Z_i]$ is finite since $Z_i \in {0, 1, -1}$. – Michael Jan 02 '23 at 23:08

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