For this problem there are 3 states zonal flow $(R1)$, intermediate $(R2)$ and blocked regime$ (R3).$ Let $A$ denote the event that more than three consecutive days are classified as blocked regime. Calculate the probability that more than three consecutive days are classified as blocked starting from day n which is already blocked, $P(A|Xn = 3)$. Assume the Markov property:
I defined the state space $S= \{{1,2,3}\}$ with$ 1$ corresponding to $R1$, $2$ corresponding to $R2 $ and $3$ corresponding to $R3$. I let $A =\{{(X_{n+1}, X_{n+2}, X_{n+3}) = R3} \}$ which i'm not sure is correct. Then $P(A|X_n=3) = P(X_{n+3}, X_{n+2}=3,X_{n+1}=3 | X_n =3)$. I am not sure if i'm right at all. Would greatly appreciate the help or reassurance
The expression should be $\mathbb{P}(X_{n+1} = 3 | X_n = 3) \cdot \mathbb{P}(X_{n+2} = 3 | X_{n+1} = 3) \cdot \mathbb{P}(X_{n+3} = 3 | X_{n+2} = 3)$