\begin{enumerate}
  \item A biased coin has probability 0.4 of showing a head. In an experiment, the coin is spun until a head appears. If a head has not appeared after 4 spins, the coin is not spun again. The random variable $X$ represents the number of times the coin is spun.
\end{enumerate}

For example, $X = 3$ if the first two spins do not show a head but the third spin does show a head. The coin would not then be spun a fourth time since the coin has already shown a head.\\
(a) Show that $\mathrm { P } ( X = 3 ) = 0.144$

The table gives some values for the probability distribution of $X$

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ &  & 0.24 & 0.144 &  \\
\hline
\end{tabular}
\end{center}

(b) (i) Write down the value of $\mathrm { P } ( X = 1 )$\\
(ii) Find $\mathrm { P } ( X = 4 )$\\
(c) Find $\mathrm { E } ( X )$\\
(d) Find $\operatorname { Var } ( X )$

The random variable $H$ represents the number of heads obtained when the coin is spun in the experiment.\\
(e) Explain why $H$ can only take the values 0 and 1 and find the probability distribution of $H$.\\
(f) Write down the value of\\
(i) $\mathrm { P } ( \{ X = 3 \} \cap \{ H = 0 \} )$\\
(ii) $\mathrm { P } ( \{ X = 4 \} \cap \{ H = 0 \} )$

The random variable $S = X + H$\\
(g) Find the probability distribution of $S$

\hfill \mbox{\textit{Edexcel S1 2017 Q6 [17]}}