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 $q_0$ and $q_1$ 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 $(1-q_0)(1-q_1)^3$.
Don't we also need to account for the probability of these digits being sent in the first place? Let $S_0$ and $S_1$ be respectively the events that $0$ and $1$ are sent, and $R_0$ and $R_1$ the events that $0$ and $1$ are received. Shouldn't the probability be
$P(S_0 \cap R_0)P(S_1 \cap R_1)^3 = P(R_0 | S_0)P(S_0)P(R_1 | S_1)^3P(S_1)^3$
$=p(1-q_0)(1-p)^3(1-q_1)^3$?
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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