A code symbol is transmitted through a communication system. The transmission pr

Question

A code symbol is transmitted through a communication system. The transmission process is not
perfect: the probability that an error occurs during transmission of the symbol is 0.15. However, if an
error occurs, there is probability 0.8 that the error is fixed upon reception.

(a) Use proper probability notation to indicate what events the two probability values indicated (0.15,
0.8) represent.
(b) What is the probability that a symbol received is the correct symbol?
(c) Let's now consider the transmission of n independent code symbols. What is the probability that k
of the n symbols received are correct?

Explanation / Answer

Let the events be S = Error occurs during transmission of the symbol T = Error is fixed upon reception a) P(S) = 0.15, P(T|S) = 0.8 Therefore 0.15 represents P(S), and 0.8 represents "T given S is true" i.e. P(T|S) b) Prob that the symbol received is correct = P(error did not occur) + P(S)*P(T|S) = 1 - P(S) + P(S)*P(T|S) = 1 - 0.15 + 0.15*0.8 = 0.97 (However if you want the prob of just error not happening, it is 0.85 only. Above I considered that if the error occurred it was fixed. That is the second term) c) Prob that k out of n are received correct = (0.97^k) * ((1-0.97)^(n-k)) (i,e, since k symbols are correct n-k are incorrect. We have to consider that prob too)

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