Q:

Probability of a fair coin's $j$th and $k$th rolls have the same outcome

A fair die rolls $n$ times, let $R_{j,k}$ be the event that the $j$th and $k$th rolls have the same outcomes. Calculate $P(R_{j,k})$ with $1\leq j,k\leq n$.
Event $R_{j,k}$ has two same outcomes for sure, then the remain $n-2$ rolls can be any number from $1$ to $6$, so $P(R_{j,k})=6\times6^{n-2}/6^n=1/6$. 
EDIT:
How to show $R_{j,k}$ are pair-wise independent.

A:

For all $j,k$ when $1\leq j<k\leq n$, then $\mathsf P(R_{j,k})=1/6$ as you ascertained.
(Note that when $j=k$, of course $\mathsf P(R_{j,j})=1$ )
To prove distinct events $R_{j,k}$ and $R_{i,h}$ are always pairwise independent, we have to show $\mathsf P(R_{j,k}, R_{i,h})=\mathsf P(R_{j,k})~\mathsf P(R_{i,h})$ for all indices $1\leq j<k\leq n$, $1\leq i<h\leq n$ in the following cases:

all indices are distinct 
two indices match, the others are distinct

