PT_bh-2-12.tex

\begin{exercise}
    \textbf{[BH.12]} Communication from Alice to Bob by sending a message (encoded in binary) across a channel.
    \begin{enumerate}
        \item First answer the original questions in your own words.
        \item Now, Alice sends $\underbrace{00\cdots0}_{n \  0's}$ to convey $0$ and $\underbrace{11\cdots1}_{n \  1's}$ to convey $1$.
        \begin{enumerate}
            \item Given that Bob receives $\underbrace{11\cdots1}_{(n-1) \ 1's}0$, what is the probability that Alice intended to convey a $1$ now?
            \item What is the probability when Alice intends to convey a $1$, Bob gets it correctly (it can happen that the message can not be decoded, e.g., half $0$'s half $1$'s)? The following code calculates the correct rate corresponding to $n=1,11,21,31,41$  (Outputs are $0.9000000$ $0.9997043$ $0.9999986$ $1.0000000$ $1.0000000$).
            \begin{minted}[]{R}
f <- function(n,k){choose(n, k)*0.9^k*0.1^(n-k)}
g <- function(n){sum(f(n, seq(floor(0.5*n)+1, n, by=1))) }
lower = 1
upper = 41
sapply(seq(lower, upper, by=10), g)
            \end{minted}

\begin{minted}{python}
import math


def f(n, k):
    return math.comb(n, k) * 0.9**k * 0.1 ** (n - k)


def g(n):
    a = [f(n, k) for k in range(math.floor(0.5 * n) + 1, n + 1)]
    return sum(a)


lower = 1
upper = 41
[print(g(i)) for i in range(lower, upper + 1, 10)]
\end{minted}
        \end{enumerate}

    \end{enumerate}
\end{exercise}