5 Kirk and Les regularly play each other at darts.
\begin{enumerate}[label=(\alph*)]
\item The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game.

Find the probability that, in a match of 15 games, Kirk wins:
\begin{enumerate}[label=(\roman*)]
\item exactly 5 games;
\item fewer than half of the games;
\item more than 2 but fewer than 7 games.
\end{enumerate}\item Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
\begin{enumerate}[label=(\roman*)]
\item Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
\item To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results:

$$\begin{array} { l l l l l l l l l l } 
8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5
\end{array}$$

Calculate the mean and standard deviation of these values.
\item Hence comment on the validity of Kirk's claim.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2006 Q5 [17]}}