| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Multi-stage with stopping condition |
| Difficulty | Moderate -0.8 This is a straightforward tree diagram problem with clearly stated probabilities and standard conditional probability calculations. The multi-stage structure is simple (max 2 attempts then one final test), requiring only basic probability multiplication and addition rules with no conceptual challenges beyond textbook exercises. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Fully correct labelled tree diagram with W1P branch probabilities: \(0.8, 0.7, 0.3\); W1F branch: \(0.2, 0.6, 0.4\); PP, PF branches after W2 with \(0.3, 0.7\) | B1 | Fully correct labelled tree diagram for each pair of branches clearly identifying written and practical, pass and fail for each intersection (no additional branches) |
| 'One written test' branch all probabilities correct | B1 | 'One written test' branch all probabilities (or %) correct |
| 'Two written tests' branch all probabilities correct, condone additional branches after W2F with probabilities 1 for PF and 0 for PP | B1 | 'Two written tests' branch all probabilities (or %) correct, condone additional branches after W2F with probabilities 1 for PF and 0 for PP |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.8 \times 0.3 + 0.2 \times 0.6 \times 0.3\) | M1 | Consistent with *their* tree diagram or correct |
| \(0.276\) or \(\frac{69}{250}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(W1 | P) = \frac{P(W1 \cap \text{Practical})}{P(\text{getting place})} = \frac{0.8 \times 0.3}{their(b)} \left[= \frac{0.24}{0.276}\right]\) | M1 |
| \(\frac{20}{23}\) or \(0.87[0]\) | A1 |
## Question 4(a):
Fully correct labelled tree diagram with W1P branch probabilities: $0.8, 0.7, 0.3$; W1F branch: $0.2, 0.6, 0.4$; PP, PF branches after W2 with $0.3, 0.7$ | B1 | Fully correct labelled tree diagram for each pair of branches clearly identifying written and practical, pass and fail for each intersection (no additional branches)
'One written test' branch all probabilities correct | B1 | 'One written test' branch all probabilities (or %) correct
'Two written tests' branch all probabilities correct, condone additional branches after W2F with probabilities 1 for PF and 0 for PP | B1 | 'Two written tests' branch all probabilities (or %) correct, condone additional branches after W2F with probabilities 1 for PF and 0 for PP
---
## Question 4(b):
$[P(W1P) \times P(PP) + P(W1F) \times P(W2P) \times P(PP)]$
$0.8 \times 0.3 + 0.2 \times 0.6 \times 0.3$ | M1 | Consistent with *their* tree diagram or correct
$0.276$ or $\frac{69}{250}$ | A1 |
---
## Question 4(c):
$P(W1|P) = \frac{P(W1 \cap \text{Practical})}{P(\text{getting place})} = \frac{0.8 \times 0.3}{their(b)} \left[= \frac{0.24}{0.276}\right]$ | M1 | Correct expression or FT *their* **(b)**
$\frac{20}{23}$ or $0.87[0]$ | A1 |
---4 To gain a place at a science college, students first have to pass a written test and then a practical test.\\
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test.
The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .
\begin{enumerate}[label=(\alph*)]
\item Draw a tree diagram to represent this information, showing the probabilities on the branches.
\item Find the probability that a randomly chosen student will succeed in gaining a place at the college.\\[0pt]
[2]
\item Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q4 [7]}}