Probability of letter transmitting and receiving

Question

One of the sequences of letters $AAAA, BBBB,CCCC$ is transmitted over a communication channel with respective probabilities $p_1,p_,p_3$ where $(p_1 + p_2 + p_3 = 1)$. the probability that each transmitted letter will be correctly understood is $\alpha$ and the probabilities that the letter will be confused with two other letters are $\frac{1}{2} (1 − \alpha)$ and $\frac{1}{2}(1 −\alpha)$. It is assumed that the letters are distorted independently. find the probability that $AAAA$ was transmitted if $ABCA$ was received.

Answer 1

I abbreviate $AAAA$, $BBBB$ etc as just $A$ and $B$.

Use Bayes' Theorem. $$ P(A \mid ABCA) = \frac{P(ABCA\mid A)P(A)}{P(ABCA)} $$ The demoninator can be expanded as $$ P(ABCA) = P(ABCA\mid A)P(A) + P(ABCA \mid B)P(B) + P(ABCA \mid C)P(C) $$