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:
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if~\(0\le X< \frac14\), my score is 4;
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if~\(\frac14\le X< \frac12\), my score is 3;
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if \(\frac12\le X< \frac34\), my score is 2;
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if \(\frac34\le X\le 1\), my score is 1.
- Show that my expected score from one dart is 15/8.
- 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:
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if my new total is greater than 3, I have lost and the game ends;
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\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.