| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Standard Bayes with discrete events |
| Difficulty | Moderate -0.8 This is a straightforward application of the law of total probability and Bayes' theorem with clearly stated probabilities and no conceptual complications. The calculations are routine: multiply and add for part (i), then apply Bayes' formula for part (ii). This is easier than average as it requires only direct formula application with no problem-solving insight. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(L) = 0.5 \times 0.04 + 0.3 \times 0.06 + 0.2 \times 0.17\) | M1 | For summing three relevant 2-factor terms |
| A1 | For correct expression | |
| \(= 0.072\ (9/25)\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B \mid L) = \frac{0.3 \times 0.06}{0.072}\) | B1 ft | For their \(0.3 \times 0.06\) in numerator, must be divided by \(k \neq 1\) |
| M1 | For dividing by their \(P(L)\) | |
| \(= 0.25\) | A1 | 3 |
## Question 3:
### Part (i)
$P(L) = 0.5 \times 0.04 + 0.3 \times 0.06 + 0.2 \times 0.17$ | M1 | For summing three relevant 2-factor terms
| A1 | For correct expression
$= 0.072\ (9/25)$ | A1 | **3** | For correct answer
### Part (ii)
$P(B \mid L) = \frac{0.3 \times 0.06}{0.072}$ | B1 ft | For their $0.3 \times 0.06$ in numerator, must be divided by $k \neq 1$
| M1 | For dividing by their $P(L)$
$= 0.25$ | A1 | **3** | For correct answer3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company $A , 30 \%$ to $B$ and $20 \%$ to $C$. A taxi from company $A$ arrives late $4 \%$ of the time, a taxi from company $B$ arrives late $6 \%$ of the time and a taxi from company $C$ arrives late $17 \%$ of the time.\\
(i) Find the probability that, when Andrea rings for a taxi, it arrives late.\\
(ii) Given that Andrea's taxi arrives late, find the conditional probability that she rang company $B$.
\hfill \mbox{\textit{CAIE S1 2004 Q3 [6]}}