| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2026 |
| Session | January |
| Marks | 11 |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Standard +0.3 This is a straightforward probability distribution question requiring: (a) summing an arithmetic series to find k, (b) adding probabilities from a given formula, and (c) using the discriminant condition b²-4ac<0. All steps are standard A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown2.04a Discrete probability distributions |
8.
The discrete random variable $R$ takes even integer values from 2 to $2 n$ inclusive.\\
The probability distribution of $R$ is given by
$$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = n ( n + 1 )$
When $n = 20$
\item find the exact value of $\mathrm { P } ( 16 \leqslant R < 26 )$
When $n = 20$, a random value $g$ of $R$ is taken and the quadratic equation in $x$
$$x ^ { 2 } + g x + 3 g = 5$$
is formed.
\item Find the exact probability that the equation has no real roots.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2026 Q8 [11]}}