| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| 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 basic counting of outcomes, simple arithmetic transformations (squaring and subtracting), and standard application of expectation and variance formulas. The die faces are explicitly given, making it purely mechanical with no problem-solving insight needed—easier than average A-level. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\) values: \(-3,\ 0,\ 5,\ 32\) | B1 | At least 3 different correct values of \(X\) (can be unsimplified) |
| Probabilities: \(\frac{1}{6},\ \frac{1}{2},\ \frac{1}{6},\ \frac{1}{6}\) | B1 | Four correct probabilities in a Probability Distribution table |
| Correct probs paired with correct values of \(X\) | B1 | Correct probs with correct values of \(X\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = -3/6 + 5/6 + 32/6 = 34/6 = 17/3\ (5.67)\) | M1 | Subst their attempts at scores in correct formula as long as 'probs' sum to 1 |
| \(\text{Var}(X) = 9/6 + 25/6 + 1024/6 - (34/6)^2\) | M1 | Subst their attempts at scores in correct var formula |
| \(= 144\left(\dfrac{1298}{9}\right)\) | A1 | Both answers correct |
| 3 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$ values: $-3,\ 0,\ 5,\ 32$ | B1 | At least 3 different correct values of $X$ (can be unsimplified) |
| Probabilities: $\frac{1}{6},\ \frac{1}{2},\ \frac{1}{6},\ \frac{1}{6}$ | B1 | Four correct probabilities in a Probability Distribution table |
| Correct probs paired with correct values of $X$ | B1 | Correct probs with correct values of $X$ |
| **Total: 3** | | |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = -3/6 + 5/6 + 32/6 = 34/6 = 17/3\ (5.67)$ | M1 | Subst their attempts at scores in correct formula as long as 'probs' sum to 1 |
| $\text{Var}(X) = 9/6 + 25/6 + 1024/6 - (34/6)^2$ | M1 | Subst their attempts at scores in correct var formula |
| $= 144\left(\dfrac{1298}{9}\right)$ | A1 | Both answers correct |
| | **3** | |
---4 A fair die with faces numbered $1,2,2,2,3,6$ is thrown. The score, $X$, is found by squaring the number on the face the die shows and then subtracting 4.\\
(i) Draw up a table to show the probability distribution of $X$.\\
(ii) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{CAIE S1 2017 Q4 [6]}}