6. The score, \(X\), for a biased spinner is given by the probability distribution
| \(x\) | 0 | 3 | 6 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 12 }\) | \(\frac { 2 } { 3 }\) | \(\frac { 1 } { 4 }\) |
Find
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
A biased coin has one face labelled 2 and the other face labelled 5 The score, \(Y\), when the coin is spun has
$$\mathrm { P } ( Y = 5 ) = p \quad \text { and } \quad \mathrm { E } ( Y ) = 3$$
- Form a linear equation in \(p\) and show that \(p = \frac { 1 } { 3 }\)
- Write down the probability distribution of \(Y\).
Sam plays a game with the spinner and the coin.
Each is spun once and Sam calculates his score, \(S\), as follows
$$\begin{aligned}
& \text { if } X = 0 \text { then } S = Y ^ { 2 }
& \text { if } X \neq 0 \text { then } S = X Y
\end{aligned}$$ - Show that \(\mathrm { P } ( S = 30 ) = \frac { 1 } { 12 }\)
- Find the probability distribution of \(S\).
- Find \(\mathrm { E } ( S )\).
Charlotte also plays the game with the spinner and the coin.
Each is spun once and Charlotte ignores the score on the coin and just uses \(X ^ { 2 }\) as her score. Sam and Charlotte each play the game a large number of times. - State, giving a reason, which of Sam and Charlotte should achieve the higher total score.