| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Independent Events |
| Type | Both independence and mutual exclusivity |
| Difficulty | Moderate -0.3 This is a straightforward probability question requiring calculation of P(S), P(T), and P(S∩T) from a simple sample space (16 outcomes), then applying standard definitions of independence and mutual exclusivity. The concepts are fundamental and the calculations routine, making it slightly easier than average, though it does require careful enumeration and understanding of two distinct probability concepts. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(S) = \frac{3}{16}\) | M1 | Sensible attempt at \(P(S)\) |
| \(P(T) = \frac{4}{16}\) | M1 | Sensible attempt at \(P(T)\) |
| \(P(S\cap T) = \frac{2}{16}\) | B1 | Correct \(P(S\cap T)\) |
| \(P(S)\times P(T) = \frac{3}{64} \neq \frac{2}{16}\) | M1 | comp \(P(S)\times P(T)\) with \(P(S\cap T)\) (their values), evaluated |
| Not independent | A1 (5) | Correct conclusion following all correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Not exclusive since \(P(S\cap T)\neq 0\); or counter example e.g. 1 and 3; or \(P(S\cup T)\neq P(S)+P(T)\) with values | B1\(\checkmark\) (1) | FT their \(P(S\cap T)\), not obtained from \(P(S)\times P(T)\), with value and statement |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(S) = \frac{3}{16}$ | M1 | Sensible attempt at $P(S)$ |
| $P(T) = \frac{4}{16}$ | M1 | Sensible attempt at $P(T)$ |
| $P(S\cap T) = \frac{2}{16}$ | B1 | Correct $P(S\cap T)$ |
| $P(S)\times P(T) = \frac{3}{64} \neq \frac{2}{16}$ | M1 | comp $P(S)\times P(T)$ with $P(S\cap T)$ (their values), evaluated |
| Not independent | A1 (5) | Correct conclusion following all correct working |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Not exclusive since $P(S\cap T)\neq 0$; or counter example e.g. 1 and 3; or $P(S\cup T)\neq P(S)+P(T)$ with values | B1$\checkmark$ (1) | FT their $P(S\cap T)$, not obtained from $P(S)\times P(T)$, with value and statement |
---3 Ellie throws two fair tetrahedral dice, each with faces numbered 1, 2, 3 and 4. She notes the numbers on the faces that the dice land on. Event $S$ is 'the sum of the two numbers is 4 '. Event $T$ is 'the product of the two numbers is an odd number'.\\
(i) Determine whether events $S$ and $T$ are independent, showing your working.\\
(ii) Are events $S$ and $T$ exclusive? Justify your answer.
\hfill \mbox{\textit{CAIE S1 2015 Q3 [6]}}