| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 6 |
| Topic | Geometric Distribution |
| Type | First success on specific trial |
| Difficulty | Moderate -0.8 This is a straightforward geometric distribution question requiring only direct formula application: (a) uses P(X=k)=(1-p)^(k-1)×p with given values, (b) recalls standard formulas E(X)=1/p and Var(X)=(1-p)/p², and (c) applies negative binomial for 2nd success. All parts are routine recall with simple arithmetic, easier than average A-level questions which typically require more problem-solving or multi-step reasoning. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
3. A football player is practising taking penalties. On each attempt the player has a $70 \%$ chance of scoring a goal. The random variable $X$ represents the number of attempts that it takes for the player to score a goal.
\begin{enumerate}[label=(\alph*)]
\item Determine $\mathrm { P } ( X = 4 )$.
\item Find each of the following.
\begin{itemize}
\item $\mathrm { E } ( X )$
\item $\operatorname { Var } ( X )$
\item Determine the probability that the player needs exactly 4 attempts to score 2 goals.\\[0pt]
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2022 Q3 [6]}}