| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a standard two-equation, two-unknown problem requiring students to use ΣP(X=x)=1 and the expectation formula E(X)=Σx·P(X=x). The algebra is straightforward with simple coefficients and no complex manipulation needed. Slightly easier than average as it's a routine textbook exercise with clear methodology. |
| Spec | 5.02b Expectation and variance: discrete random variables |
| \(x\) | - 2 | - 1 | 0.5 | 1 | 2 |
| \(\mathrm { P } ( X = x )\) | 0.12 | \(p\) | \(q\) | 0.16 | 0.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.12 + p + q + 0.16 + 0.3 = 1\) | B1 | Sum of probabilities \(= 1\); \(p + q = 0.42\) OE |
| \(-0.24 - p + 0.5q + 0.16 + 0.6 = 0.28\) | B1 | Form equation using \(E(X) = 0.28\); \(-p + 0.5q = -0.24\) OE; Accept unsimplified |
| Attempt to solve *their* two equations in \(p\) and \(q\) | M1 | Either substitution method to form a single equation in either \(p\) or \(q\) and finding values for both unknowns. Or elimination method by writing both equations in the same form (usually \(ap + bq = c\)) and \(+\) or \(-\) to find an equation in one unknown and finding values for both unknowns |
| \(q = 0.12,\ p = 0.3\) | A1 | CAO, both WWW. If M0 awarded SC B1 for both correct WWW |
| 4 |
## Question 1:
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.12 + p + q + 0.16 + 0.3 = 1$ | **B1** | Sum of probabilities $= 1$; $p + q = 0.42$ OE |
| $-0.24 - p + 0.5q + 0.16 + 0.6 = 0.28$ | **B1** | Form equation using $E(X) = 0.28$; $-p + 0.5q = -0.24$ OE; Accept unsimplified |
| Attempt to solve *their* two equations in $p$ and $q$ | **M1** | **Either** substitution method to form a single equation in either $p$ or $q$ and finding values for both unknowns. **Or** elimination method by writing both equations in the same form (usually $ap + bq = c$) and $+$ or $-$ to find an equation in one unknown and finding values for both unknowns |
| $q = 0.12,\ p = 0.3$ | **A1** | CAO, both WWW. If M0 awarded **SC B1** for both correct WWW |
| | **4** | |1 The probability distribution table for a random variable $X$ is shown below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 2 & - 1 & 0.5 & 1 & 2 \\
\hline
$\mathrm { P } ( X = x )$ & 0.12 & $p$ & $q$ & 0.16 & 0.3 \\
\hline
\end{tabular}
\end{center}
Given that $\mathrm { E } ( X ) = 0.28$, find the value of $p$ and the value of $q$.\\
\hfill \mbox{\textit{CAIE S1 2022 Q1 [4]}}