For a random permutation of length $n=100$, Let $p_{12}$ be the probability that the values $1$ and $2$ are in the same cycle. For a permutation $v \in S_n$, a cycle of length $k$ is a vector of integers $(a_1, a_2, a_3, a_4, .., a_k)$ such that $v(a_1) = a_2, v(a_2) = a_3, .., v(a_{k-1})=a_k$ and $v(a_k) = 1$. For example, for the permutation $v = (3,4,1,5,2)$ we have the cycle $(1,3)$ of length $2$ because $v(1) = 3$ and $v(3) = 1$. We also have the cycle $(2,4,5)$ of length $3$, because $v(2) = 4, v(4) = 5$ and $v(5)=2$. See also another example of the cycles of a permutation in the figure above. <br> Let $X_k$ be the number of cycles of length $k$ in a random permutation of length $n$ selected uniformly from $S_n$. Estimate this probability using $10,000$ simulations. Denote your estimator by $\hat{p}_{12}$. Estimate also the standard deviation of your estimator, i.e. provide and estimate $\hat{\sigma}(\hat{p}_{12})$ of $\sigma(\hat{p}_{12})$.

y <- c(1,2,3,4,5)
y