| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 7 |
| Topic | Combinations & Selection |
| Type | Probability of specific committee composition |
| Difficulty | Standard +0.8 Part (a) requires careful interpretation of conditional probability with combinations (selecting 3 white from 8 given 5 remain), which is non-trivial. Part (b) is particularly challenging as it requires treating 'exactly three green counters together' systematically—students must consider arrangements where 3 greens form a block while the 4th is separate, avoiding overcounting. Both parts demand problem-solving beyond routine application of formulas, typical of Further Statistics questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks |
|---|---|
| M1, A1, A1 | (Product of two \(^nC_r\)) ÷ \(^nC_r\) Or At least two \(^nC_r\) correct; Any exact form or awrt 0.279; \(\frac{8}{28} \times \frac{7}{27} \times \frac{6}{26} \times \frac{20}{25} \times \ldots \times \frac{16}{21} = 0.27934\ldots\) |
| Answer | Marks |
|---|---|
| M1, A1, M1, A1 | Or e.g. find \(_{12}C_1 - (\text{all separate}) +\#(\text{all together}) + \#(2,1,1) \times 3 + \#(2,2))\); Divide by \(_{12}C_4\) oe; (no additional guidance) |
## (a)
$^5C_1 \times ^{20}C_5$
$\frac{28}{C_8}$
$\frac{56 \times 15504}{3108105} = 0.27934\ldots$
| M1, A1, A1 | (Product of two $^nC_r$) ÷ $^nC_r$ Or At least two $^nC_r$ correct; Any exact form or awrt 0.279; $\frac{8}{28} \times \frac{7}{27} \times \frac{6}{26} \times \frac{20}{25} \times \ldots \times \frac{16}{21} = 0.27934\ldots$ |
## (b)
x B x B x B x B x B x B x B x B x
GGG in one x, G in another: 9×8
$÷ \frac{12!}{8! \times 4!}$
$= \frac{72}{495} = \frac{8}{55}$ or 0.145...
| M1, A1, M1, A1 | Or e.g. find $_{12}C_1 - (\text{all separate}) +\#(\text{all together}) + \#(2,1,1) \times 3 + \#(2,2))$; Divide by $_{12}C_4$ oe; (no additional guidance) |
---3
\begin{enumerate}[label=(\alph*)]
\item Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
\item Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q3 [7]}}