Probability in a Communication Channel

In a binary communication system, the symbols 0 and 1 are sent through a channel, but noise can switch 1 to 0 or vice versa with certain probabilities. The error can be estimated by Bayes' rule.
Let be the event of sending 0 (represented by ) and be the event of sending 1 (represented by ), with corresponding probabilities and . On the receiving side, and are the events of receiving 0 or 1. Represented by and , is the probability of receiving 1 given that 0 was sent, is the probability of receiving 1 given that 1 was sent, is the conditional probability of given , and so on.

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Reference
[1] P. Z. Peebles, Jr., Probability, Random Variables and Random Signal Principles, 4th ed., New York: McGraw-Hill, 2000.
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