We throw an fair dice $n$ times. Let $S_n$ be the number of throws with even number of dots on the dice.
1.) Calculate the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n)$$
2.) Express the value of the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n + \sqrt{n})$$ as a definite integral of suitable function.
First can we say $$S_n = X_1+X_2+...+X_n$$ $$X_1+X_2+...+X_n \sim \mathrm{Geo}( \frac{1}{2} )$$
I know that probability of having even dots show up on the dice is $\frac{1}{2}$. So if we throw the dice $n$ times does that mean that the limit of $S_n$ will converge toward $\frac{n}{2}$? And if so can we than say
$$S_n = \frac{n}{2}$$ $$ n = 2S_n$$
I’m not very good at solving this type of exercise so any help would be much appreciated.
A sum of independent bernoulli random variables (which you've noted each have probability $\frac12$) will be a binomial random variable, but in this case more importantly the resulting sum's cumulative distribution function will tend to that of the normal distribution with the same mean and variance, by the central limit theorem. That should answer both questions simultaneously.