| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Find unknown probability parameter |
| Difficulty | Moderate -0.8 This is a straightforward tree diagram problem requiring basic probability laws. Part (i) is routine diagram drawing, part (ii) involves solving a simple linear equation using the law of total probability (0.82x + 0.18(0.9) = 0.285), and part (iii) applies Bayes' theorem with values already computed. All steps are standard textbook exercises with no novel insight required. |
| Spec | 2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Tree diagram with E branch: \(0.82\), probabilities \(x\) (GNS) and \(1-x\) (Not GNS); F branch: \(0.18\), probabilities \(0.9\) (GNS) and \(0.1\) (Not GNS) | B1 | Must see at least 4 probs correct including one with \(x\) in, correct shape |
| B1 | Shape, clear labels/annotation and all probs correct | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.82x + 0.18 \times 0.9 = 0.285\) | M1 | Equation with \(x\) in, two 2-factors on one side |
| \(x = 0.15\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(E \mid notGNS) = \frac{P(E \cap notGNS)}{P(notGNS)}\) | M1 | Attempt at \(P(E \cap \text{not GNS})\) seen as num or denom of fraction |
| M1 | Attempt at \(P(\text{not GNS})\) seen anywhere | |
| \(= \frac{0.82 \times 0.85}{1 - 0.285} = 0.975\) | A1 | Correct answer |
| 3 |
## Question 5:
### Part 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Tree diagram with E branch: $0.82$, probabilities $x$ (GNS) and $1-x$ (Not GNS); F branch: $0.18$, probabilities $0.9$ (GNS) and $0.1$ (Not GNS) | B1 | Must see at least 4 probs correct including one with $x$ in, correct shape |
| | B1 | Shape, clear labels/annotation and all probs correct |
| | **2** | |
### Part 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.82x + 0.18 \times 0.9 = 0.285$ | M1 | Equation with $x$ in, two 2-factors on one side |
| $x = 0.15$ | A1 | |
| | **2** | |
### Part 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(E \mid notGNS) = \frac{P(E \cap notGNS)}{P(notGNS)}$ | M1 | Attempt at $P(E \cap \text{not GNS})$ seen as num or denom of fraction |
| | M1 | Attempt at $P(\text{not GNS})$ seen anywhere |
| $= \frac{0.82 \times 0.85}{1 - 0.285} = 0.975$ | A1 | Correct answer |
| | **3** | |
---5 Over a period of time Julian finds that on long-distance flights he flies economy class on $82 \%$ of flights. On the rest of the flights he flies first class. When he flies economy class, the probability that he gets a good night's sleep is $x$. When he flies first class, the probability that he gets a good night's sleep is 0.9 .\\
(i) Draw a fully labelled tree diagram to illustrate this situation.
The probability that Julian gets a good night's sleep on a randomly chosen flight is 0.285 .\\
(ii) Find the value of $x$.\\
(iii) Given that on a particular flight Julian does not get a good night's sleep, find the probability that he is flying economy class.\\
\hfill \mbox{\textit{CAIE S1 2017 Q5 [7]}}