| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 9 |
| Topic | Hypergeometric Distribution |
| Type | Probability with multiple categories |
| Difficulty | Standard +0.8 This is a Further Maths statistics question requiring understanding of hypergeometric distributions across multiple stages with conditional probability. Part (i) is straightforward application of combinations, but part (ii) requires careful tracking of remaining cards after set A is selected and applying the hypergeometric distribution twice in sequence—this multi-stage conditional reasoning with changing populations elevates it above typical A-level questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{^{10}C_3 \times \,^{40}C_2}{^{50}C_3}\) | M1 | At least one correct "\(C_r\)" seen |
| A1 | Correct expression | |
| \(= \frac{2340}{52969}\) or 0.044176... | A1 | Exact or awrt 0.0442 |
| Answer | Marks | Guidance |
|---|---|---|
| \(1/5! = \frac{1}{120}\) | B1 | Any equivalent form, allow awrt 0.00833 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{^7C_2 \times \,^{33}C_6 \times \,^2C_1 \times \,^2C_1}{^{40}C_{10}} \times \frac{^{30}C_{10}}{^{30}C_{10}}\) | M1 | Correct structure for \(B\) |
| A1 | Correct numbers for \(B\) | |
| M1 | Two terms of this sort multiplied | |
| A1 | Fully correct expression | |
| \(= \frac{450}{8177}\) or 0.055032... | A1 | Exact or awrt 0.0550 |
## Part (i)(a)
$\frac{^{10}C_3 \times \,^{40}C_2}{^{50}C_3}$ | M1 | At least one correct "$C_r$" seen
| A1 | Correct expression
$= \frac{2340}{52969}$ or 0.044176... | A1 | Exact or awrt 0.0442 | Or probability method: $\frac{120 \times 780}{2118760}$
## Part (i)(b)
$1/5! = \frac{1}{120}$ | B1 | Any equivalent form, allow awrt 0.00833
## Part (ii)
$\frac{^7C_2 \times \,^{33}C_6 \times \,^2C_1 \times \,^2C_1}{^{40}C_{10}} \times \frac{^{30}C_{10}}{^{30}C_{10}}$ | M1 | Correct structure for $B$
| A1 | Correct numbers for $B$
| M1 | Two terms of this sort multiplied
| A1 | Fully correct expression
$= \frac{450}{8177}$ or 0.055032... | A1 | Exact or awrt 0.05503 Adila has a pack of 50 cards.\\
(i) Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
\item Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order.
10 of the 50 cards are blue and the rest are green.\\
(ii) Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled $A , B$ and $C$. Given that $A$ contains 3 blue cards and 7 green cards, find the probability that $B$ contains exactly 2 blue cards and $C$ contains exactly 3 blue cards.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q3 [9]}}