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 :
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
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 $P(Y_3=a)$