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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.

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closed as off-topic by Parcly Taxel, Davide Giraudo, Stefan Mesken, Daniel W. Farlow, erfink May 28 '17 at 20:46

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    $\begingroup$ Have you tried to solve it yourself? Where did you get stuck? $\endgroup$ – Mickey May 28 '17 at 11:59
  • $\begingroup$ I don't understand how to approach solving this problem so I need someone to explain me $\endgroup$ – Nemanja Petrovic May 28 '17 at 12:02
  • $\begingroup$ If you open an arbitrary probability theory book, you can find questions with similar constructions, so from those you can have at least a starting point, which means you don't need for someone to explain to you how to approach to these kind of questions.After that, if you cannot solve it, you can come and ask for help about the point that you have stuck with. $\endgroup$ – onurcanbektas May 28 '17 at 14:31
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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 $$ P(ABCA) = P(ABCA\mid A)P(A) + P(ABCA \mid B)P(B) + P(ABCA \mid C)P(C) $$

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