A source transmits a string of symbols through a channel. Each symbol is 0 or 1 with probability p and 1−p respectively, and is received incorrectly with probability q0 and q1 respectively. Errors in different symbol transmissions are independent. What is the probability that the string 1011 is received correctly?
The given answer is :
By independence, the probability that the string 1011 is received correctly is (1−q0)(1−q1)3.
Don't we also need to account for the probability of these digits being sent in the first place? Let S0 and S1 be respectively the events that 0 and 1 are sent, and R0 and R1 the events that 0 and 1 are received. Shouldn't the probability be
P(S0∩R0)P(S1∩R1)3=P(R0|S0)P(S0)P(R1|S1)3P(S1)3
=p(1−q0)(1−p)3(1−q1)3?
Thanks
Answer
The bit probabilities p and 1−p are irrelevant to the problem, which only concerns how often a fixed string 1011 is received correctly, not how likely that string forms in the first place.
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