| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring only routine techniques: summing probabilities to find k, constructing a table, then applying standard variance formula. All steps are mechanical with no problem-solving insight needed, making it easier than average but not trivial due to the arithmetic involved. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Probs \(6k, 3k, 2k, 6k, 11k\) so \(28k = 1\), giving \(k = \frac{1}{28}\) | B1 | \(k\) must be identified |
| Table: \(x = -2, -1, 0, 2, 3\) with \(P(X=x) = \frac{6}{28}, \frac{3}{28}, \frac{2}{28}, \frac{6}{28}, \frac{11}{28}\) (decimals: \(0.2143, 0.1071, 0.07143, 0.2143, 0.3929\)) | M1 | Table with correct outcomes and 2 correct probabilities. FT substituting their \(k\) correctly into formula, \(0 < p < 1\). No additional \(x\) values unless probability 0. Condone in terms of \(k\) of form \(\frac{6k}{28}\) or \(6k\) |
| A1 | Fully correct. Decimal answers to at least 3 sig figures, condone not summing exactly to 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = \left[-2 \times \frac{6}{28} + -1 \times \frac{3}{28} + 0 \times \frac{2}{28} + 2 \times \frac{6}{28} + 3 \times \frac{11}{28}\right]\) \(= \frac{1}{28}(-12-3+12+33) = \frac{15}{14}\) | M1 | Accept unsimplified expression. May be calculated in variance. FT their table with 5 probabilities \(0 < p < 1\) that sum to 1 |
| \(\text{Var}(X) = \frac{6\times(-2)^2 + 3\times(-1)^2 + 6\times 2^2 + 11\times 3^2}{28} - \left(\frac{15}{14}\right)^2\) | M1 | Appropriate variance formula using their \((E(X))^2\) value. FT their table with at least 4 probabilities \(0 < p < 1\), that may not sum to 1 |
| \(= 4.21,\ 4\frac{41}{196}\) | A1 | Condone \(\frac{825}{196}\). If one or both M marks not awarded, SC B1 for correct answer WWW |
# Question 2:
## Part 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Probs $6k, 3k, 2k, 6k, 11k$ so $28k = 1$, giving $k = \frac{1}{28}$ | B1 | $k$ must be identified |
| Table: $x = -2, -1, 0, 2, 3$ with $P(X=x) = \frac{6}{28}, \frac{3}{28}, \frac{2}{28}, \frac{6}{28}, \frac{11}{28}$ (decimals: $0.2143, 0.1071, 0.07143, 0.2143, 0.3929$) | M1 | Table with correct outcomes and 2 correct probabilities. FT substituting their $k$ correctly into formula, $0 < p < 1$. No additional $x$ values unless probability 0. Condone in terms of $k$ of form $\frac{6k}{28}$ or $6k$ |
| | A1 | Fully correct. Decimal answers to at least 3 sig figures, condone not summing exactly to 1 |
## Part 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = \left[-2 \times \frac{6}{28} + -1 \times \frac{3}{28} + 0 \times \frac{2}{28} + 2 \times \frac{6}{28} + 3 \times \frac{11}{28}\right]$ $= \frac{1}{28}(-12-3+12+33) = \frac{15}{14}$ | M1 | Accept unsimplified expression. May be calculated in variance. FT their table with 5 probabilities $0 < p < 1$ that sum to 1 |
| $\text{Var}(X) = \frac{6\times(-2)^2 + 3\times(-1)^2 + 6\times 2^2 + 11\times 3^2}{28} - \left(\frac{15}{14}\right)^2$ | M1 | Appropriate variance formula using their $(E(X))^2$ value. FT their table with at least 4 probabilities $0 < p < 1$, that may not sum to 1 |
| $= 4.21,\ 4\frac{41}{196}$ | A1 | Condone $\frac{825}{196}$. If one or both M marks not awarded, **SC B1** for correct answer WWW |
---2 The random variable $X$ takes the values $- 2 , - 1,0,2,3$. It is given that $\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.
\item Find the value of $\operatorname { Var } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q2 [6]}}