| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Conditional with three or more stages |
| Difficulty | Standard +0.3 This is a multi-stage conditional probability problem requiring careful tracking of pocket contents across two transfers. Part (i) is straightforward multiplication, part (ii) requires considering multiple pathways to achieve X=1, and part (iii) involves Bayes' theorem. While it has three parts and requires systematic case analysis, the individual probability calculations are routine and the tree diagram structure is standard for S1 level. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks |
|---|---|
| M1 | Multiplying two different probs |
| A1 | [2] |
| Answer | Marks |
|---|---|
| M1 | Finding \(P(R,R) = \frac{3}{5}\) |
| Answer | Marks |
|---|---|
| M1 | Summing two options |
| A1 | [3] |
| Answer | Marks |
|---|---|
| M1 | Their (i) seen as numerator or denominator of a fraction |
| M1 | \(\frac{3}{4} \times p_1 + \frac{1}{4} \times p_2\) seen anywhere |
| A1 | \(\frac{1}{4}\) (unsimplified) seen as numerator or denominator of a fraction |
| A1 | Working with explanation [4] |
**(i)**
$P(B, B) = \frac{1}{4} \times \frac{2}{5} = \frac{1}{10}$
M1 | Multiplying two different probs
A1 | [2]
**(ii)**
$P(X = 1) = P(R,R) + P(B,B)$
M1 | Finding $P(R,R) = \frac{3}{5}$
$= \frac{3}{4} \times \frac{4}{5} + \frac{1}{10} = \frac{14}{20}$ or $\frac{7}{10}$
M1 | Summing two options
A1 | [3]
**(iii)**
$P(B \mid B) = \frac{P(B \cap B)}{P(B)} = \frac{1/10}{\frac{3}{4} \times \frac{1}{5} + \frac{1}{4} \times \frac{2}{5}} = \frac{2}{5}$
M1 | Their (i) seen as numerator or denominator of a fraction
M1 | $\frac{3}{4} \times p_1 + \frac{1}{4} \times p_2$ seen anywhere
A1 | $\frac{1}{4}$ (unsimplified) seen as numerator or denominator of a fraction
A1 | Working with explanation [4]
---6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation $T$ ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.\\
(i) Find the probability that, when Deeti carries out operation $T$, she takes a blue pen from her left pocket and then a blue pen from her right pocket.
The random variable $X$ is the number of blue pens in Deeti's left pocket after carrying out operation $T$.\\
(ii) Find $\mathrm { P } ( X = 1 )$.\\
(iii) Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.
\hfill \mbox{\textit{CAIE S1 2016 Q6 [9]}}