Problem $2$ Eight equally likely messages are encoded using the following set of codewords for

transmission over a binary symmetric channel $($BSC$)$ with crossover probability $$:

x$1=0000$ x$5=1001$

x$2=0011$ x$6=1010$

x$3=0101$ x$7=1100$

x$4=0110$ x$8=1111$

If the sequence $0000$ is received at the channel output, find:

a$)$ The amount of info provided about x$1$ by the first digit received.

b$)$ The additional amounts of info about x$1$ provided by the second, the third, and the

fourth received digit.