Moderate -0.8 This is a straightforward S2 question testing standard expectation and variance calculations with discrete distributions. All parts involve direct application of formulas (E(X) = Σxp(x), properties of linear transformations, independence) with no conceptual challenges or problem-solving required. The most demanding aspect is part (c)(ii) requiring enumeration of cases, but this is routine bookwork for S2 level.
A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by
\(\boldsymbol { x }\)
1
2
3
4
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)
0.2
0.1
0.4
0.3
Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained.
Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by
$$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3 \\ \frac { 7 } { 10 } & y = 4 \end{cases}$$
The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 .
Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
Calculate:
\(\mathrm { P } ( X > 1 )\);
\(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
\(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).
4
\begin{enumerate}[label=(\alph*)]
\item A red biased tetrahedral die is rolled. The number, $X$, on the face on which it lands has the probability distribution given by
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 1 & 2 & 3 & 4 \\
\hline
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$ & 0.2 & 0.1 & 0.4 & 0.3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
\item The red die is now rolled three times. The random variable $S$ is the sum of the three numbers obtained.
Find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$.
\end{enumerate}\item A blue biased tetrahedral die is rolled. The number, $Y$, on the face on which it lands has the probability distribution given by
$$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3 \\ \frac { 7 } { 10 } & y = 4 \end{cases}$$
The random variable $T$ is the value obtained when the number on the face on which it lands is multiplied by 3 .
Calculate $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
\item Calculate:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( X > 1 )$;
\item $\mathrm { P } ( X + T \leqslant 9$ and $X > 1 )$;
\item $\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2011 Q4 [18]}}