| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.8 This is a straightforward two-equation, two-unknown problem using basic probability axioms (probabilities sum to 1) and expectation formula. It requires only routine algebraic manipulation with no conceptual difficulty or problem-solving insight, making it easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 6 |
| \(\mathrm { P } ( X = x )\) | 0.15 | \(p\) | 0.4 | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p + q = 0.45\) | M1 | Equation involving \(\Sigma P(x) = 1\) |
| \(0.15 + 2p + 1.2 + 6q = 3.05\) | M1 | Equation using \(E(X) = 3.05\) |
| \(q = 0.2\) | M1 | Solving simultaneous equations to one variable |
| \(p = 0.25\) | A1 | Both answers correct |
| Total: 4 |
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p + q = 0.45$ | M1 | Equation involving $\Sigma P(x) = 1$ |
| $0.15 + 2p + 1.2 + 6q = 3.05$ | M1 | Equation using $E(X) = 3.05$ |
| $q = 0.2$ | M1 | Solving simultaneous equations to one variable |
| $p = 0.25$ | A1 | Both answers correct |
| **Total: 4** | | |
---1 The discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 6 \\
\hline
$\mathrm { P } ( X = x )$ & 0.15 & $p$ & 0.4 & $q$ \\
\hline
\end{tabular}
\end{center}
Given that $\mathrm { E } ( X ) = 3.05$, find the values of $p$ and $q$.\\
\hfill \mbox{\textit{CAIE S1 2017 Q1 [4]}}