Conditional Probability - Transmitter sending signals of $3$ letters $a, b, c$ I'm learning conditional probability, specifically Bayes' theorem, and need help with the following problem :

A transmitter is sending signals that consist of $3$ letters $a, b, c$ with frequency of $30$%, $20$% and $50$%, respectively. The receiver receives the letters successively with the same order sent. The receptions are mutually independent. The signals are received with errors that where estimated as follows :

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*When the transmitter sends the letter $a$, the receiver receives the letter $a$ with probability $70$%, the letter $b$ with probability $20$% and the letter $c$ with probability $10$%;


*When the transmitter sends the letter $b$, the receiver receives the letter $a$ with probability $30$%, the letter $b$ with probability $60$% and the letter $c$ with probability $10$%;


*When the transmitter sends the letter $c$, the receiver receives the letter $a$ with probability $10$%, the letter $b$ with probability $50$% and the letter $c$ with probability $40$%;
$(1)$ What is the probability that the receiver receives the signal $(bba)$, if the transmitter send the signal $(bba)$?
$(2)$ What is the probability the receiver receives the signal $(bba)$
$(3)$ What is the probability the transmitter sent the signal $(abc)$ if the receiver received the signal $(bba)$.

Since I'm new to this subject, I like solving these problems using tree diagrams. I had some difficulties setting up the tree diagram for this problem. I'm not sure if I made the good diagram or not. Here it is :

Is this the adequate tree diagram for this problem? It is unclear to me how I should use this diagram to answer questions $(1)$, $(2)$ and $(3)$. Questions $(1)$ and $(3)$ are clearly related to Bayes' theorem since we have conditional probabilities but I don't know how to set up the appropriate events. For instance in $(1)$, writting $P((bba) | (bba))$ for "the probability that the receiver receives the signal $(bba)$, if the transmitter send the signal $(bba)$" does not seem to make a lot of sense. Question $(2)$ is also tricky since the are many paths leading to the signal $(bba)$. I also don't understand what it means for the receptions to be mutually independent and if this sentence is of any importance to this problem. Any help would be greatly appreciated
 A: You can define elementary events in order to manage this exercise.
$X_j=i$: The letter i is sent at position j, with $i\in\{a,b,c\}$ and $j\in\{1,2,3\}$
$Y_j=i$: The letter i is received at position j, with $i\in\{a,b,c\}$ and $j\in\{1,2,3\}$

Then in $(1)$ it is asked for $P(Y_1=b,Y_2=b,Y_3=a|X_1=b,X_2=b,X_3=a)$
Since the receptions are  mutually independent this is equal to
$P(Y_1=b|X_1=b)\cdot P(Y_2=b|X_2=b)\cdot P(Y_3=a|X_3=a)=0.6\cdot 0.6\cdot 0.4$
This numbers can be read off at your diagram.
The independency in this context means that the probability to receive the letter $i$ at position $j$ does not depend on what the reception on another position is. Formally it is $P(Y_j=i|Y_k=m)=P(Y_j=i) \ \forall \ \ j\neq k$

In $(2)$ you calculate first the probability that the receiver reveive $b$ at the beginning: $P(Y_1=b)$
Here you can use the total law of probability:
$$=P(Y_1=b|X_1=a)\cdot P(X_1=a)+P(Y_1=b|X_1=b)\cdot P(X_1=b)+P(Y_1=b|X_1=c)\cdot P(X_1=c)$$
$=0.2\cdot 0.3+0.6\cdot 0.2+0.5\cdot 0.5$
These numbers can be read off your diagram as well. Similar calculations for $P(Y_2=b)$ and $P(Y_3=a)$. The sum of the three pobabilities then is $P(Y_1=b \cap Y_2=b \cap Y_3=a)$

In $(3)$ it is asked for $$P(X_1=a|Y_1=b\cap X_2=b|Y_2=b\cap X_3=c|Y_3=a)=P(X_1=a|Y_1=b)\cdot P( X_2=b|Y_2=b)\cdot P(X_3=c|Y_3=a)$$
Using the Bayes Theorem you obtain $P(X_1=a|Y_1=b)=\frac{P(Y_1=b|X_1=a)\cdot P(X_1=a)}{P(Y_1=b)}$
Similar calculations for $P( X_2=b|Y_2=b)$ and $P(X_3=c|Y_3=a)$. You can use the results in $(2)$ for $P(Y_1=b), P(Y_2=b)$ and $P(Y_3=a)$ 
