| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring systematic enumeration of outcomes (6×6=36 equally likely cases), calculating products, and finding probabilities by counting. Part (b) is direct application of E(X) formula. Requires careful organization but no conceptual difficulty or problem-solving insight—below average difficulty for A-level. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\): 1, 2, 3, 4, 6 with \(P(X=x)\): \(\frac{6}{36}, \frac{12}{36}, \frac{6}{36}, \frac{6}{36}, \frac{6}{36}\) (i.e. \(\frac{1}{6}, \frac{1}{3}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}\)) or decimals 0.167, 0.333, 0.167, 0.167, 0.167 | B1 | Table with correct \(x\) values and at least one correct probability linked with the correct \(x\)-value. Values need not be in order, lines may not be drawn, may be vertical, \(x\) and \(P(X)\) may be omitted. Condone any additional \(x\) values if probability stated as 0 |
| B1 | 4 correct probabilities linked with correct \(x\)-values, need not be in table, accept unsimplified | |
| B1 | 5 correct probabilities linked with correct \(x\)-values, may not be in table. Decimals correct to at least 3 SF. SC B1 4 or 5 probabilities summing to 1 placed in a probability distribution table with 4 or 5 \(x\)-values between 1 and 6 inclusive | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([\text{E}(X) = \frac{1}{36}(6 + 24 + 18 + 24 + 36) =]\ 3\) | B1 FT | FT *their* table with 4 or 5 probabilities (\(0 < p < 1\)) summing to 1 |
| 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$: 1, 2, 3, 4, 6 with $P(X=x)$: $\frac{6}{36}, \frac{12}{36}, \frac{6}{36}, \frac{6}{36}, \frac{6}{36}$ (i.e. $\frac{1}{6}, \frac{1}{3}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}$) or decimals 0.167, 0.333, 0.167, 0.167, 0.167 | **B1** | Table with correct $x$ values and at least one correct probability linked with the correct $x$-value. Values need not be in order, lines may not be drawn, may be vertical, $x$ and $P(X)$ may be omitted. Condone any additional $x$ values if probability stated as 0 |
| | **B1** | 4 correct probabilities linked with correct $x$-values, need not be in table, accept unsimplified |
| | **B1** | 5 correct probabilities linked with correct $x$-values, may not be in table. Decimals correct to at least 3 SF. **SC B1** 4 or 5 probabilities summing to 1 placed in a probability distribution table with 4 or 5 $x$-values between 1 and 6 inclusive |
| | **3** | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\text{E}(X) = \frac{1}{36}(6 + 24 + 18 + 24 + 36) =]\ 3$ | **B1 FT** | FT *their* table with 4 or 5 probabilities ($0 < p < 1$) summing to 1 |
| | **1** | |
---2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled $1,1,2,2,3,3$. Both dice are thrown. The random variable $X$ is the product of the scores on the two dice.
\begin{enumerate}[label=(\alph*)]
\item Draw up the probability distribution table for $X$.
\item Find $\mathrm { E } ( X )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q2 [4]}}