| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2021 |
| Session | February |
| Marks | 10 |
| Topic | Discrete Probability Distributions |
| Type | Sum or product of two independent values |
| Difficulty | Standard +0.3 This is a straightforward statistics question requiring basic probability concepts. Part (a) uses the fact that probabilities sum to 1 (routine algebra with fractions). Part (b) requires listing cases where two independent variables sum to 80 and multiplying probabilities (systematic but mechanical). Part (c) combines probability with arithmetic sequences and quadrilateral angle properties, requiring more steps but still using standard A-level techniques without novel insight. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(d\) | 10 | 20 | 30 | 40 | 50 |
| \(P(D = d)\) | \(\frac{k}{10}\) | \(\frac{k}{20}\) | \(\frac{k}{30}\) | \(\frac{k}{40}\) | \(\frac{k}{50}\) |
The discrete random variable $D$ has the following probability distribution
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$d$ & 10 & 20 & 30 & 40 & 50 \\
\hline
$P(D = d)$ & $\frac{k}{10}$ & $\frac{k}{20}$ & $\frac{k}{30}$ & $\frac{k}{40}$ & $\frac{k}{50}$ \\
\hline
\end{tabular}
\end{center}
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that the value of $k$ is $\frac{600}{137}$
[2]
\item The random variables $D_1$ and $D_2$ are independent and each have the same distribution as $D$.
Find $P(D_1 + D_2 = 80)$
Give your answer to 3 significant figures.
[3]
\item A single observation of $D$ is made.
The value obtained, $d$, is the common difference of an arithmetic sequence.
The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral $Q$
Find the exact probability that the smallest angle of $Q$ is more than $50°$
[5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2021 Q6 [10]}}