A radio wants to send a symbol, which is equally likely to be a or b, over a communication link. The symbols are sometimes decoded incorrectly. Specially, if a is sent, it is decoded as b with probability 1/3 (and is otherwise correct). If b is sent, it is decoded as a with probability 1/4 (and is otherwise correct). To overcome these errors, the transmitter first selects a symbol to transmit where A is the event that it selects a and B is the event that is selects b. It will then repeat this same symbol m times over the communication link. Let Ai be the event that the ith received symbol is a and Bi that the ith received symbol is b. You may assume that the events Al; ...;Am; B1; .;Bm are conditionally independent given the event A or the event B. This implies many decompositions of the probabilities of intersecting events, such as m P[A1 N A2 n... n Am|A] = II
P[A;|A] i=1 P[B1 n A2 n ... n Bm|A] =
P[B1|A]P[A2|A] P[Bm|A]
P[B1 n A2 n... n Bm|B] = P[B1\B]P[A2|B] .. . P[Bm|B]
(a) Calculate P[Ai].
(b) Are the events A1 and A2 independent?
(c) What is the probability that the true symbol is a given that the first received symbol is a and the second is b? (e. g. PĻA|Ai and B2]).
(d) What is the probability that the true symbol is a given that the first three received symbols are a? (e. g. P[A|A1 and A2 and A3])
(e) Consider the following error correction scheme. The transmitter sends the symbol 3 times. If the receiver sees 2 or more of the same symbol, it takes that symbol as its guess for the true symbol. Hence, the receiver is decoding using majority voting. What is the probability the guess is correct?