A roulette player sets $n$ times to $0$; the random variable $X_i$ denotes the result of the $i$-th game.
a) Specify the distribution of each $X_i$.
b) Argue why the relative frequency of successes converges stochastically to the theoretical probability of success. What does this result mean in practice?
c) Determine the necessary number $n$ of games, so that the relative frequency of the achievements $\overline{X}_n=\frac{1}{n}\sum_{i=1}^nX_i$ is with a probability of at least $90\%$ in the interval $\frac{1}{37}-\frac{1}{74}<\overline{X}_n<\frac{1}{37}+\frac{1}{74}$.
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I have done the following:
a) Each $X_i$ describes succes or faillure, therefore each $X_i$ is bernoulli distributed.
b) Does this have a relation with the law of large numbers?
c) According to the central limit theorem we have that $\overline{X}$ approximates the normal distribution for large $n$ with the parameters \begin{equation*}E(\overline{X})=\frac{1}{2}=0.5 \ \text{ and } \ V(\overline{X})=\frac{\sigma_X^2}{n}=\frac{\left (\frac{1}{4}\right )^2}{n}=\frac{1}{16n}\Rightarrow \sigma_{\overline{X}}=\frac{1}{4\sqrt{n}}\end{equation*}
The relative frequency of achievements $\overline{X}_n$ should be with probability of at least $90\%$ in the bounds $\frac{1}{37}-\frac{1}{74}<\overline{X}_n<\frac{1}{37}+\frac{1}{74} \Rightarrow \frac{1}{74}<\overline{X}_n<\frac{3}{74}$.
So we have the following:
\begin{align*}P\left (\frac{1}{74}<\overline{X}_n<\frac{3}{74}\right )\geq 90\% &\Rightarrow \Phi \left (\frac{\frac{3}{74}-0.5}{\frac{1}{4\sqrt{n}}}\right )-\Phi \left (\frac{\frac{1}{74}-0.5}{\frac{1}{4\sqrt{n}}}\right )\geq 0.9 \\ & \Rightarrow \Phi \left (-1.83784 \sqrt{n}\right )-\Phi \left (-1.94595 \sqrt{n}\right )\geq 0.9 \\ & \Rightarrow 1-\Phi \left (1.83784 \sqrt{n}\right )-1+\Phi \left (1.94595 \sqrt{n}\right )\geq 0.9 \\ & \Rightarrow -\Phi \left (1.83784 \sqrt{n}\right )+\Phi \left (1.94595 \sqrt{n}\right )\geq 0.9\end{align*}
Is everything so far? If yes, how could we calculate the last expression where we have two values of the distribution function?
For a) you should be more specific with the distribution of $X_i$. It is the result of one spin of the wheel and is either $0$ or $1$. What is the probability of $0$? of $1$?
For b) yes. You should look at the normal approximation and claim that the fraction of successes converges to $\frac 1{37}$ assuming you are playing with only a single $0$.
For c) $E(\overline X) \neq \frac 12$. If I could find a roulette wheel where my probability of success was $\frac 12$ I would be betting a lot there an be rich. Your $\sigma_X$ is also badly wrong. You have a discrete distribution for $X_i$ with $P(X_i=0)=\frac {36}{37}$