How to approach the probability problem about the 4 liars I'm currently taking High School probability classes, and one of the problems in our text book is the following problem: 
There are four people sitting in a row, and they have to transport a message of the type "YES" or "NO" by telling to each other. Namely, the first person has to tell the message to the second one, the second one to the third one, the third one to the fourth one, and the fourth one is going to tell it further more. However, each of them tells the truth with probability of 1/3. If we know that the last person said the correct information (the message the first person received) what is the probability that the first person told the second one the correct information?
I'm not sure how to define the events here and how to approach this problem. I started by trying to explore each different case, since there are 16 of them (2 for each of them, to lie or tell the truth). However, I'm not sure how to finish this problem.
Please give me some hints or help on how to approach this problem.
Edit: I got a result 13/81, is there a way to confirm my calculations?
 A: Some hints about how to proceed. You said: 

I started by trying to explore each different case, since there are 16 of them (2 for each of them, to lie or tell the truth).

This is a great strategy. Hopefully in your exploration you have included the probability of each case as well -- for instance, the probability that all four players tell the truth is $(2/3)^4$, and the probability that all four players lie is $(1/3)^4$.
One useful way to organize this information, if you haven't already, is to construct a tree diagram. Your tree should have four "levels" (consisting of the choice of each player to lie or not) and 16 ends. Here, the ends correspond to a full collection of the 4 choices of each player. Write the probability of each end of the tree; verify that your total probability is $1$ before proceeding.
From here: identify the ends of the tree that correspond to the situation at hand (i.e. the fourth player delivering the correct message). Cross out all the other ends; they're irrelevant now. The remaining branches are your new probability space. Figure out the total probability remaining, then figure out the proportion of the remaining space that corresponds to the first person telling the truth.
(The binomial / Bayes comment is fine too, of course, but I was supposing you may not have seen that language at a high school level.)

Regarding your comment (and noting that the probability of lying is actually $2/3$): your answer of $13/81$ is still incorrect, but you're on the right track.
If event $A$ is $\{\text{person 1 gives the correct information to person 2}\}$ and event $B$ is $\{\text{person 4 ultimately gives the correct information}\}$, then $13/81$ correctly computes $\mathbb P(A \cap B)$. However, this isn't what you want; you want $\mathbb P (A \mid B)$, which is $\mathbb P(A \cap B) / \mathbb P(B)$. So, you're partway there; now, compute $\mathbb P(B)$ (which you can again use your tree diagram to do) and divide by it. Don't forget to consider the case where all 4 players lie in part B!
