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\section*{Written Assignment 12}

\begin{question}{1}
Solve book problem 23.7.
\end{question}

\clearpage

\begin{question}{2}
In the word game Scrabble, players end their turns by picking letter tiles to replace the ones they have just used. Suppose a player picks 4 letter tiles from the 7 tiles that remain: \\
A, B, C, D, E, F, and G.
\begin{enumerate}[label=(\alph*)]
\item How many different selections of 4 letters are possible?
\item How many different â€œwordsâ€ (sequences of letters, not necessarily valid
English words) can be made from 4 of these 7 letters?
\end{enumerate}
\end{question}

\begin{question}{3}
Solve book problem 26.1 (b).
\end{question}

\begin{question}{4}
Solve book problem 26.2. Show your calculations.
\end{question}

\clearpage

\begin{question}{5}
An error-correcting code is a technique for signaling and correcting errors when sending a message over an unreliable communication channel. One simple example is a repetition code, in which the message is repeated a fixed number of times. If any bit differs between repetitions, it is known to be corrupted (and the true value is assumed to be the value that appears most frequently).

Suppose that for a given communication channel, the probability that any given bit is corrupted is 0.001, and the outcome for each bit is independent. As an error-correcting code, messages are repeated 3 times.
\begin{enumerate}[label=(\alph*)]
\item Suppose a message of length 20 is sent (for a total of 60 bits after repetition). What is the probability that at least one bit is corrupted?
\item What is the maximum-length message that can be sent such that the probability of at least one bit being corrupted is less than 0.5?
\item (Optional) The weakness of a repetition code is that if a given bit is corrupted in every repetition, there is no signal of the error. Consider a message of length 8,000,000 bits (i.e., 1~gigabyte). What is the probability that there exists a bit that is corrupted in all 3 repetitions of the message?
\end{enumerate}
\end{question}

\end{document}
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