| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Independent events test |
| Difficulty | Standard +0.3 This is a straightforward probability question requiring systematic enumeration of outcomes from two dice throws. Part (i) and (ii) involve counting favorable outcomes from 64 total possibilities, while part (iii) tests understanding of independence using P(R∩S) = P(R)×P(S). The calculations are routine with no conceptual subtleties, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(R)\ [(1,4),(2,5),(3,6),(4,7),(5,8)] \times \frac{2}{64} = \frac{10}{64}\) | M1, A1 (2) | List of at least 4 different options or possibility space diagram; correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(S) = [(3,8)(3,7)(4,8)(4,7)(4,6)(4,5)(5,8)\ (5,7)(5,6)(6,8)(6,7)(7,8)] \times 2 + (5,5)(6,6)(7,7)(8,8) = \frac{28}{64}\) | M1, A1 (2) | List of at least 14 different options or ticks from possibility space; correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(R \cap S) = \frac{4}{64}\); \(\frac{4}{64} \neq \frac{10}{64} \times \frac{28}{64}\); Events are not independent | B1, M1, A1 (3) | Comparing their \(P(R \cap S)\) with (i) \(\times\) (ii) with values; correct answer |
## Question 3:
**(i)**
$P(R)\ [(1,4),(2,5),(3,6),(4,7),(5,8)] \times \frac{2}{64} = \frac{10}{64}$ | M1, A1 (2) | List of at least 4 different options or possibility space diagram; correct answer
**(ii)**
$P(S) = [(3,8)(3,7)(4,8)(4,7)(4,6)(4,5)(5,8)\ (5,7)(5,6)(6,8)(6,7)(7,8)] \times 2 + (5,5)(6,6)(7,7)(8,8) = \frac{28}{64}$ | M1, A1 (2) | List of at least 14 different options or ticks from possibility space; correct answer
**(iii)**
$P(R \cap S) = \frac{4}{64}$; $\frac{4}{64} \neq \frac{10}{64} \times \frac{28}{64}$; Events are not independent | B1, M1, A1 (3) | Comparing their $P(R \cap S)$ with (i) $\times$ (ii) with values; correct answer
---3 A fair eight-sided die has faces marked $1,2,3,4,5,6,7,8$. The score when the die is thrown is the number on the face the die lands on. The die is thrown twice.
\begin{itemize}
\item Event $R$ is 'one of the scores is exactly 3 greater than the other score'.
\item Event $S$ is 'the product of the scores is more than 19'.\\
(i) Find the probability of $R$.\\
(ii) Find the probability of $S$.\\
(iii) Determine whether events $R$ and $S$ are independent. Justify your answer.
\end{itemize}
\hfill \mbox{\textit{CAIE S1 2016 Q3 [7]}}