Exercise 1.38 (A Game Of Chairs)

Question

Tyrion, Cersei, and ten other people are sitting at a round table, with their seating arrangement having been randomly assigned. What is the probability that Tyrion and Cersei are sitting next to each other? Find this in two ways:
\begin{enumerate}[label=(\alph*)]
\item using a sample space of size $12!$, where an outcome is fully detailed about the seating;
\item using a much smaller sample space, which focuses on Tyrion and Cersei.
\end{enumerate}

fifthist x · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item  There are $12$ choices of seats for Tyron and Cersei so that they sit 
next to each other ($11$ cases, where they take $i-1$ and $i$ positions and 
1 case, where they take 1 and 12th position because the table is round). 
Tyron can sit to the left or to the right of Cersei.  
The remaining $10$ people can be ordered in $10!$ ways, so the answer is
$$\frac{24	imes10!}{12!} = \frac{2}{11}$$

\item There are $\binom{12}{2}$ choices of seats to be assigned to Tyron and 
Cersei, but only $12$ choices where they sit next to each other. 
Since every assignment of seats is equally likely the answer is
$$\frac{12}{\binom{12}{2}} = \frac{2}{11}$$
\end{enumerate}

\begin{figure}
\includegraphics[width=0.5	extwidth]{lannister.png}
\end{figure}

Exercise 1.38 (A Game Of Chairs)

Answers

Comments

Exercise 1.38 (A Game Of Chairs)

Answers

Comments

Add answer