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A data source generates bits $0$ or $1$ independently with probabilities $0.4$ and $0.6$ respectively. The data transmitted via a communication channel is corrupted by noise. Each bit may be received in error with probability $0.2$. Given that the $2$-bit data sequence $01$ is received, determine the probability distribution of the transmitted data sequence.

What i tried

$P$(of getting $0$)=$0.4+(0.6)*(0.2)$

$P$(of getting $1$)=$0.6+(0.4)*(0.2)$

Then i multiply both the probabilities together to get the probability of getting the $2$=bit data $01$. Am i correct? COuld anyone explain. Thanks

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I will give you this hint. If we got a zero, we must understand this means we faithfully received a zero or we unfaithfully received a 1. So:

$P(0) = P(0~ \mathrm{and~ no~ error}) + P(1~ \mathrm{and~ error}) = 0.8 \cdot 0.4 + 0.2\cdot 0.6 = 0.44$

Hopefully this helps you finish the problem.

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