| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2024 |
| Session | January |
| Marks | 11 |
| Topic | Discrete Probability Distributions |
| Type | Comparison or ordering of two independent values |
| Difficulty | Standard +0.8 Part (a) is routine probability axiom application, parts (b-c) require systematic enumeration of cases with independent observations. Part (d) is substantially harder, requiring careful combinatorial reasoning about paths to exactly 7 in exactly 5 steps without overshooting—this demands strategic case analysis and is non-standard for A-level, pushing into olympiad-style problem-solving territory. |
| Spec | 2.03a Mutually exclusive and independent events2.04a Discrete probability distributions |
The probability distribution of a random variable $X$ is modelled as follows.
$$\text{P}(X = x) = \begin{cases}
\frac{k}{x} & x = 1, 2, 3, 4, \\
0 & \text{otherwise,}
\end{cases}$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac{12}{25}$. [2]
\item Show in a table the values of $X$ and their probabilities. [1]
\item The values of three independent observations of $X$ are denoted by $X_1$, $X_2$ and $X_3$.
Find P$(X_1 > X_2 + X_3)$. [3]
\end{enumerate}
In a game, a player notes the values of successive independent observations of $X$ and keeps a running total. The aim of the game is to reach a total of exactly 7.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Determine the probability that a total of exactly 7 is first reached on the 5th observation. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2024 Q7 [11]}}