Let each one of the individuals $A_{0}, A_{1}, A_{2}$ and $A_{3}$ say the truth with probability $p$ independentely from each other and that $A_{3}$ confirms that $A_{2}$ denies that $A_{1}$ says that $A_{0}$ is lying. We ask for the probability that $A_{0}$ is saying the truth.
We model this problem using this stochastic process: Let $X_{0}=1$ or $0$ if $A_{0}$ is saying the truth or is lying, respectively. Let $X_{n}=1$ or $0$ if what $A_{n}$ is saying implies that $A_{0}$ is saying the truth or is lying, respectively.
I have the following questions:
$(1)$ Why $\{ X_{n} : n\in{\mathbb{N}} \}$ is a Markov chain ?
$(2)$ We need to search for the probabilities of first transition. If $p_{ij}=P(X_{1}=j | X_{0}=i)$, why is it $p_{00}=p_{11}=p$ and $p_{10}=p_{01}=1-p$ ?
Here are my takes. Let me know where I am being mistaken and correct me.
$(1)$ $X_n$ is a Markov chain because knowing if what $A_{n}$ says, implies whether $A_{0}$ is saying the truth or no is sufficient to make predictions on whether what $A_{n+1}$ says, implies if $A_{0}$ says the truth or not, due to the hypothesis.
$(2)$ I think I can understand $p_{00}, p_{01}$ and $p_{10}$. My problem is on $p_{11}$ which could also indicate wrong understanding of the others as well. So, for $P(X_{1}=1|X_{0}=1)$ we have that, given that $A_{0}$ says the truth, what is the probability that what $A_{1}$ says, implies that $A_{0}$ is saying the truth. Now, since $A_{1}$ says that $A_{0}$ is lying, given that $A_{0}$ says the truth, in order to imply that $A_{0}$ is saying the truth, $A_{1}$ has to be lying. So we ask for the probability that he is lying which gotta be equal to $1-p$ or $p_{11}=1-p$. Where's the mistake ?
Thank you in advance.