Problem source

When Septimus Moneybags throws darts at a dart board they are certain
	to end on the board (a disc of radius $a$) but, it must be admitted,
	otherwise are uniformly randomly distributed over the board. 
	\begin{questionparts}
	\item Show that the distance $R$ that his shot lands from the centre of
	the board is a random variable with variance $a^{2}/18.$
	
	\item At a charity fete he can buy $m$ throws for $\pounds(12+m)$, but
	he must choose $m$ before he starts to throw. If at least one of
	his throws lands with $a/\sqrt{10}$ of the centre he wins back $\pounds 12$.
	In order to show that a good sport he is, he is determined to play
	but, being a careful man, he wishes to choose $m$ so as to minimise
	his expected loss. What values of $m$ should he choose? 
\end{questionparts}

Solution source

\begin{questionparts}
\item $\,$ \begin{align*}
&& \mathbb{P}(R < d) &= \frac{\pi d^2}{\pi a^2} \\
&&&= \frac{d^2}{a^2} \\
\Rightarrow && f_R(d) &= \frac{2d}{a^2}\\
\\
&& \E[R] &= \int_0^a x \cdot f_R(x) \d x \\
&&&= \int_0^a \frac{2x^2}{a^2} \d x \\
&&&= \frac{2a}{3} \\
\\
&& \E[R^2] &= \int_0^a x^2 \cdot f_R(x) \d x \\
&&&= \int_0^a \frac{2x^3}{a^2} \d x \\
&&&= \frac{a^2}{2} \\
\Rightarrow && \var[R] &= \frac{a^2}2  - \frac{4a^2}{9} \\
&&7= \frac{a^2}{18}
\end{align*}
\item Let $p = \mathbb{P}(R < \frac{a}{\sqrt{10}}) = \frac{a^2}{10a^2} = \frac{1}{10}$ be the probability of hitting the target on each throw.

His expected loss is $(12+m)p^m + m(1-p^m) = 12p^m + m$. 

\begin{array}{c|c}
m & \text{expected loss} \\ \hline 
0 & 12 \\
1 & \frac{12}{10} + 1 \approx 2.2 \\
2 & \frac{12}{100} + 2 \approx 2.12 \\
\end{array}
If he takes more than $2$ throws it will definitely cost more than $3$, therefore he should take exactly $2$ throws.
\end{questionparts}