| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Conditional probability in arrangements |
| Difficulty | Standard +0.8 This is a multi-part permutations question that progresses from standard arrangements with repeated letters (part a), through conditional arrangements (part b), to conditional probability requiring careful counting of restricted arrangements (part c). The final part demands understanding of conditional probability P(A|B) = P(A∩B)/P(B), treating letter groups as single units, and managing multiple constraints simultaneously—significantly above average difficulty but within reach of well-prepared S1 students. |
| Spec | 2.03a Mutually exclusive and independent events5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\frac{8!}{2!3!} =\right] 3360\) | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{6!}{2!2!}\) | M1 | \(\frac{6!}{2!f!}\); \(f = 1, 2, 3\) |
| \(180\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[P(OOO\ | CC) = \frac{P(OOO \cap CC)}{P(CC)} =\right]\) \(\frac{5!}{\dfrac{7!}{3!}}\) | M1 |
| M1 | \(\frac{h}{\dfrac{7!}{3!}}\) or \(\frac{h}{\dfrac{8!}{3!}}\), where \(h\) is a positive integer. Condone division by 3360 in denominator | |
| \(= \frac{120}{840} = \frac{1}{7} = 0.143\) | A1 | \(0.1428571\ldots\) to at least 3SF. If M0 scored SC B1 for \(\frac{1}{7}\) WWW |
| 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{8!}{2!3!} =\right] 3360$ | B1 | |
| | **1** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{6!}{2!2!}$ | M1 | $\frac{6!}{2!f!}$; $f = 1, 2, 3$ |
| $180$ | A1 | |
| | **2** | |
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[P(OOO\|CC) = \frac{P(OOO \cap CC)}{P(CC)} =\right]$ $\frac{5!}{\dfrac{7!}{3!}}$ | M1 | $\frac{5!}{g}$, $g$ a positive integer, $g \neq 3360, 1$. Condone numerator of $\frac{5!}{3360g}$ |
| | M1 | $\frac{h}{\dfrac{7!}{3!}}$ or $\frac{h}{\dfrac{8!}{3!}}$, where $h$ is a positive integer. Condone division by 3360 in denominator |
| $= \frac{120}{840} = \frac{1}{7} = 0.143$ | A1 | $0.1428571\ldots$ to at least 3SF. If M0 scored **SC B1** for $\frac{1}{7}$ WWW |
| | **3** | |
---3
\begin{enumerate}[label=(\alph*)]
\item Find the number of different arrangements of the 8 letters in the word COCOONED.
\item Find the number of different arrangements of the 8 letters in the word COCOONED in which the first letter is O and the last letter is N .
\item Find the probability that a randomly chosen arrangement of the 8 letters in the word COCOONED has all three Os together given that the two Cs are next to each other.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q3 [6]}}