Equilibrium Probability in a Markov Chain [Closed]

I'm guessing you are referring to what is more commonly known as a stationary distribution. Suppose you had a Markov chain $X_t$ for $t>0$. After it runs for a long time, you could take a look at its state, e.g. $X_{1000}$. What would that state be? Well, we don't know for sure, but there's a chance it's equal to $A$, another chance it's $B$, etc., i.e. it follows some random distribution probability. This is the stationary distribution, and is a vector of probabilities that sum to 1.

The point of a stationary distribution is that it's a limiting distribution; as $t$ tends to infinity, the distribution of states you might find yourself in (without knowing the immediate prior state) tends to some limit.

However, there is a caveat: the stationary distribution doesn't always exist.