Problem source

When I throw a dart at a target, the probability that it lands a 
distance $X$ from the centre is a random variable with density
function
\[
\mathrm{f}(x)=\begin{cases}
2x & \text{ if }0\leqslant x\leqslant1;\\
0 & \text{ otherwise.}
\end{cases}
\]
I score points according to the position of the dart as follows:
%\begin{center}
%\begin{tabular}{c|c}
%Range of $X$ & my score \\[1mm]
%\hline\\
%$0\le X< \frac14$ & 4 \\[2mm]
%$\frac14\le X< \frac12$ & 3 \\[2mm]
%$\frac12\le X< \frac34$ & 2 \\[2mm]
%$\frac34\le X\le 1$ & 1 
%\end{tabular}
%\end{center}
%\newline\hspace*{10mm} 
if~$0\le X< \frac14$, my score is 4;  
%\newline\hspace*{10mm} 
if~$\frac14\le X< \frac12$, my score is  3;
%\newline\hspace*{10mm} 
if $\frac12\le X< \frac34$, my score is  2;
%\newline\hspace*{10mm} 
if $\frac34\le X\le 1$, my score is  1. 
\begin{questionparts}
\item Show that my expected score from one dart is 15/8.
\item I play a game with the following rules. 
I start off with a total score 0, and each time~I throw a dart
my score on that throw is added to my total. Then:
\newline
\hspace*{10mm} 
if my new total is greater than 3, I have lost and the game ends;
\newline
\hspace*{10mm} if my new total is 3, I have won and the game ends;
\newline
\hspace*{10mm} if my new total is less than 3, I throw again.
 Show that, if I
have won such a game, the probability that I threw the dart three
times is 343/2231.
\end{questionparts}