| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypergeometric Distribution |
| Type | Calculate variance given expectation |
| Difficulty | Standard +0.3 This is a straightforward hypergeometric distribution question requiring standard probability calculations and variance formula application. Part (i) is routine verification, part (ii) involves computing 3-4 probabilities using combinations, and part (iii) requires calculating E(X²) from the distribution table then using Var(X) = E(X²) - [E(X)]². All steps are mechanical applications of standard S1 techniques with no conceptual challenges or novel insights required. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(\text{exactly 2}) = \frac{^6C_2}{^8C_4} = \frac{15}{70} = \frac{3}{14}\) AG | M1 | \(^6C_x / ^8C_x\) seen or \(^6C_2\) mult by 4 fractions (last 2 can be implied) |
| OR \(P(2) = \frac{6}{8} \times \frac{5}{7} \times \frac{2}{6} \times \frac{1}{5} \times ^4C_2 = \frac{3}{14}\) AG | A1 | 2 |
| (ii) | \(x\) | 2 |
| Prob | 3/14 | 8/14 |
| B1 | one correct prob other than P(2) | |
| B1♦ | 3 | third correct prob if \(\sum = 1\) |
| (iii) \(\text{Var}(X) = \frac{12}{14} + \frac{72}{14} + \frac{48}{14} - 3^2\) | M1 | using \(\sum x^2 p - 3^2\) (or their {\(E(X)\)}\(^2\)) must be evaluated |
| \(= \frac{3}{7}\) (0.429) | A1 | 2 |
(i) $P(\text{exactly 2}) = \frac{^6C_2}{^8C_4} = \frac{15}{70} = \frac{3}{14}$ AG | M1 | $^6C_x / ^8C_x$ seen or $^6C_2$ mult by 4 fractions (last 2 can be implied)
OR $P(2) = \frac{6}{8} \times \frac{5}{7} \times \frac{2}{6} \times \frac{1}{5} \times ^4C_2 = \frac{3}{14}$ AG | A1 | 2 | Answer legit obtained
(ii) | $x$ | 2 | 3 | 4 |
| Prob | 3/14 | 8/14 | 3/14 | | B1 | 2, 3, 4 only in top line
| | | | | B1 | one correct prob other than P(2)
| | | | | B1♦ | 3 | third correct prob if $\sum = 1$
(iii) $\text{Var}(X) = \frac{12}{14} + \frac{72}{14} + \frac{48}{14} - 3^2$ | M1 | using $\sum x^2 p - 3^2$ (or their {$E(X)$}$^2$) must be evaluated
$= \frac{3}{7}$ (0.429) | A1 | 2 | correct answer4 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.\\
(i) Show that the probability that she chooses exactly 2 paperback books is $\frac { 3 } { 14 }$.\\
(ii) Draw up the probability distribution table for $X$.\\
(iii) You are given that $\mathrm { E } ( X ) = 3$. Find $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{CAIE S1 2014 Q4 [7]}}