The Markov Chain is uniformly distributed across all $50$x$50$ matrices of entries $0$ and $$1 with no neighboring $1's$. I am supposed to run a MC simulation to check the probability that the $25$x$25$ entry of this matrix is a $1$.
My method is to start with the zeroes matrix and run the chain by randomly selecting an entry to change. If the new matrix is in the chain, a counter variable increases by one. If the $25$x$25$ entry is a $1$, another counter increases. I was very careful with the chain.
I ran this method $2,000,000$ times and got that the probability was $.49$. This seems a little high, and I'm worried that the chain doesn't touch anywhere near the $2$^$50$ entries. Is this method OK?