Standard +0.3 This is a standard S2 discrete probability distribution question requiring routine calculations of expectation, variance, and conditional probability. While part (b)(i) requires systematic enumeration of cases and part (b)(iii) uses conditional probability, these are textbook techniques with no novel insight needed. The multi-part structure and 18 total marks indicate moderate length, but each component is straightforward application of formulas.
The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
\(r\)
\(\leqslant 2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(\geqslant 9\)
\(\mathrm{P}(R = r)\)
\(0\)
\(0.1\)
\(0.2\)
\(0.3\)
\(0.25\)
\(0.1\)
\(0.05\)
\(0\)
Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
Calculate \(\mathrm{E}(R)\). [2 marks]
Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
\(s\)
\(\leqslant 2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(\geqslant 9\)
\(\mathrm{P}(S = s)\)
\(0\)
\(0.15\)
\(0.4\)
\(0.3\)
\(0.1\)
\(0.03\)
\(0.02\)
\(0\)
Assuming that \(R\) and \(S\) are independent:
show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
calculate the probability that, when \(5\) members are selected at random, at least \(4\) of them complete the first two holes in fewer than \(9\) strokes; [3 marks]
calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]
\begin{enumerate}[label=(\alph*)]
\item The number of strokes, $R$, taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$r$ & $\leqslant 2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $\geqslant 9$ \\
\hline
$\mathrm{P}(R = r)$ & $0$ & $0.1$ & $0.2$ & $0.3$ & $0.25$ & $0.1$ & $0.05$ & $0$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Determine the probability that a member, selected at random, takes at least $5$ strokes to complete the first hole. [1 mark]
\item Calculate $\mathrm{E}(R)$. [2 marks]
\item Show that $\mathrm{Var}(R) = 1.66$. [4 marks]
\end{enumerate}
\item The number of strokes, $S$, taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$s$ & $\leqslant 2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $\geqslant 9$ \\
\hline
$\mathrm{P}(S = s)$ & $0$ & $0.15$ & $0.4$ & $0.3$ & $0.1$ & $0.03$ & $0.02$ & $0$ \\
\hline
\end{tabular}
\end{center}
Assuming that $R$ and $S$ are independent:
\begin{enumerate}[label=(\roman*)]
\item show that $\mathrm{P}(R + S \leqslant 8) = 0.24$; [5 marks]
\item calculate the probability that, when $5$ members are selected at random, at least $4$ of them complete the first two holes in fewer than $9$ strokes; [3 marks]
\item calculate $\mathrm{P}(R = 4 \mid R + S \leqslant 8)$. [3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2010 Q6 [18]}}