| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Find unknown probability parameter |
| Difficulty | Moderate -0.3 This is a standard tree diagram problem requiring setting up an equation from given probabilities and solving for x, then applying conditional probability. The conceptual framework is straightforward (total probability and Bayes' theorem), though it requires careful algebraic manipulation. Slightly easier than average due to its routine structure and clear setup. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1-x)\) and \(0.45\) (or \(0.3\)) | B1 | Seen, either on tree diagram or elsewhere |
| Beginners: \(0.7 \times x + 0.45 \times (1-x) = 0.5\) or Advanced: \(0.3 \times x + 0.55 \times (1-x) = 0.5\) | M1 | One of the three correct probability equations |
| \(x = 0.2\) oe | A1 | Correct answer |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(M \mid A) = \frac{P(M \cap A)}{P(A)} = \frac{0.2 \times 0.3}{0.5}\) | M1 | \(`i' \times 0.3\) as num or denom of a fraction |
| M1 | \(0.5\) (or \((1-`i') \times 0.55 + `i' \times 0.3\) unsimplified) seen as denom of a fraction | |
| \(= 0.12 \left(\frac{3}{25}\right)\) | A1 | Correct answer |
| Total: 3 |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1-x)$ and $0.45$ (or $0.3$) | B1 | Seen, either on tree diagram or elsewhere |
| Beginners: $0.7 \times x + 0.45 \times (1-x) = 0.5$ or Advanced: $0.3 \times x + 0.55 \times (1-x) = 0.5$ | M1 | One of the three correct probability equations |
| $x = 0.2$ oe | A1 | Correct answer |
| **Total: 3** | | |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(M \mid A) = \frac{P(M \cap A)}{P(A)} = \frac{0.2 \times 0.3}{0.5}$ | M1 | $`i' \times 0.3$ as num or denom of a fraction |
| | M1 | $0.5$ (or $(1-`i') \times 0.55 + `i' \times 0.3$ unsimplified) seen as denom of a fraction |
| $= 0.12 \left(\frac{3}{25}\right)$ | A1 | Correct answer |
| **Total: 3** | | |3 The members of a swimming club are classified either as 'Advanced swimmers' or 'Beginners'. The proportion of members who are male is $x$, and the proportion of males who are Beginners is 0.7 . The proportion of females who are Advanced swimmers is 0.55 . This information is shown in the tree diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-04_435_974_482_587}
For a randomly chosen member, the probability of being an Advanced swimmer is the same as the probability of being a Beginner.\\
(i) Find $x$.\\
(ii) Given that a randomly chosen member is an Advanced swimmer, find the probability that the member is male.\\
\hfill \mbox{\textit{CAIE S1 2018 Q3 [6]}}