| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from table |
| Difficulty | Moderate -0.5 This is a straightforward variance calculation from a probability distribution table that students have already constructed. Once the table is drawn in part (b) and E(X) is given, finding Var(X) requires only the standard formula Var(X) = E(X²) - [E(X)]², involving basic arithmetic with small numbers. This is a routine textbook exercise testing recall of the variance formula rather than problem-solving. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER Solution 1: \(P(\text{exactly } 2) = \frac{^6C_2 \cdot ^2C_2}{^8C_4}\) | (M1) | \(\frac{^6C_x}{^8C_y}\) seen |
| \(= \frac{15}{70} = \frac{3}{14}\) | A1) | AG CWO |
| OR Solution 2: \(P(2) = \frac{6}{8} \times \frac{5}{7} \times \frac{2}{6} \times {}^4C_2\) | (M1) | \({}^4C_x\) multiplied by 4 fractions |
| \(= \frac{3}{14}\) | A1) | AG CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\): 2, 3, 4 | B1 | 2, 3, 4 only in top line |
| Prob: \(\frac{3}{14}\), \(\frac{8}{14}\), \(\frac{3}{14}\) | B1 | one correct probability other than \(P(2)\) |
| (third row correct) | B1FT | third correct probability \(FT\ \Sigma = 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(X) = \frac{12}{14} + \frac{72}{14} + \frac{48}{14} - 3^2\) | M1 | using \(\Sigma x^2 p - 3^2\) |
| \(= \frac{3}{7} = 0.429\) | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER** Solution 1: $P(\text{exactly } 2) = \frac{^6C_2 \cdot ^2C_2}{^8C_4}$ | (M1) | $\frac{^6C_x}{^8C_y}$ seen |
| $= \frac{15}{70} = \frac{3}{14}$ | A1) | AG CWO |
| **OR** Solution 2: $P(2) = \frac{6}{8} \times \frac{5}{7} \times \frac{2}{6} \times {}^4C_2$ | (M1) | ${}^4C_x$ multiplied by 4 fractions |
| $= \frac{3}{14}$ | A1) | AG CWO |
**Total: 2 marks**
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$: 2, 3, 4 | B1 | 2, 3, 4 only in top line |
| Prob: $\frac{3}{14}$, $\frac{8}{14}$, $\frac{3}{14}$ | B1 | one correct probability other than $P(2)$ |
| (third row correct) | B1FT | third correct probability $FT\ \Sigma = 1$ |
**Total: 3 marks**
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(X) = \frac{12}{14} + \frac{72}{14} + \frac{48}{14} - 3^2$ | M1 | using $\Sigma x^2 p - 3^2$ |
| $= \frac{3}{7} = 0.429$ | A1 | |
**Total: 2 marks**
---3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable $X$ represents the number of paperback books she chooses.\\
(a) Show that the probability that she chooses exactly 2 paperback books is $\frac { 3 } { 14 }$.\\
(b) Draw up the probability distribution table for $X$.\\
(c) You are given that $\mathrm { E } ( X ) = 3$.
Find $\operatorname { Var } ( X )$.\\
\hfill \mbox{\textit{CAIE S1 2020 Q3 [7]}}