| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Multi-stage game or match outcomes |
| Difficulty | Standard +0.8 This question requires constructing a conditional probability tree diagram, applying Bayes' theorem for part (b) with P(exactly 1 success | at least 1 success), and careful probability calculations for mutually exclusive compound events in part (c). The state-dependent probabilities and multi-stage reasoning elevate this above routine conditional probability exercises, though the techniques are standard S1 material. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Tree diagram with 1st, 2nd, 3rd jumps; S branch 0.2, F branch 0.8 first jump; S branches 0.3, F branches 0.7 second jump; S branches 0.1/0.3, F branches 0.9/0.7 third jump | B1 | First and second jumps correct with probabilities and outcomes identified |
| Third jump correct with probabilities and outcomes identified | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| FFS: \(0.8 \times 0.9 \times 0.1 = 0.072\) | M1 | Two or three correct 3-factor probabilities added, correct or FT from part 6(a). Accept unsimplified. |
| [Total = probability of 1 success =] \(0.254\ \left(\frac{127}{500}\right)\) | A1 | Accept unsimplified. |
| [Probability of at least 1 success \(= 1 - 0.8 \times 0.9 \times 0.9 =\)] \(0.352\ \left(\frac{44}{125}\right)\) | B1 FT | Accept unsimplified. |
| \(P(\text{exactly 1 success} \mid \text{at least 1 success}) = \frac{\text{their } 0.254}{\text{their } 0.352}\) | M1 | Accept unsimplified. |
| \(0.722,\ \frac{127}{176}\) | A1 | \(0.7215 < p \leqslant 0.722\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.8 \times 0.9 \times 0.1 \times 0.3 \times 0.3 = 0.005832\) [FFFSSS] | M1 | \(a \times b \times c \times d \times e \times f\) FT from *their* tree diagram. Either \(a\), \(b\) and \(c\) all \(= 0.8\) or \(0.9\) (at least one of each) and \(d\), \(e\) and \(f\) all \(= 0.1\) or \(0.3\) (at least one of each). Or \(a\), \(b\), \(c = 0.2\) or \(0.3\) (at least one of each) and \(d\), \(e\), \(f = 0.7\) or \(0.9\) (at least one of each). |
| \(0.2 \times 0.3 \times 0.3 \times 0.7 \times 0.9 \times 0.9 = 0.010206\) [SSSFFF] | A1 | Either correct. Accept unsimplified. |
| [Total \(=\)] \(0.0160[38]\) | A1 |
## Question 6:
**Part 6(a):**
Tree diagram with 1st, 2nd, 3rd jumps; S branch 0.2, F branch 0.8 first jump; S branches 0.3, F branches 0.7 second jump; S branches 0.1/0.3, F branches 0.9/0.7 third jump | B1 | First and second jumps correct with probabilities and outcomes identified
Third jump correct with probabilities and outcomes identified | B1 |
**Part 6(b):**
SFF: $0.2 \times 0.7 \times 0.9 = 0.126$
FSF: $0.8 \times 0.1 \times 0.7 = 0.056$
FFS: $0.8 \times 0.9 \times 0.1 = 0.072$ | M1 | Two or three correct 3-factor probabilities added, correct or FT from part 6(a). Accept unsimplified.
[Total = probability of 1 success =] $0.254\ \left(\frac{127}{500}\right)$ | A1 | Accept unsimplified.
[Probability of at least 1 success $= 1 - 0.8 \times 0.9 \times 0.9 =$] $0.352\ \left(\frac{44}{125}\right)$ | B1 FT | Accept unsimplified.
$P(\text{exactly 1 success} \mid \text{at least 1 success}) = \frac{\text{their } 0.254}{\text{their } 0.352}$ | M1 | Accept unsimplified.
$0.722,\ \frac{127}{176}$ | A1 | $0.7215 < p \leqslant 0.722$
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.8 \times 0.9 \times 0.1 \times 0.3 \times 0.3 = 0.005832$ [FFFSSS] | M1 | $a \times b \times c \times d \times e \times f$ FT from *their* tree diagram. Either $a$, $b$ and $c$ all $= 0.8$ or $0.9$ (at least one of each) and $d$, $e$ and $f$ all $= 0.1$ or $0.3$ (at least one of each). Or $a$, $b$, $c = 0.2$ or $0.3$ (at least one of each) and $d$, $e$, $f = 0.7$ or $0.9$ (at least one of each). |
| $0.2 \times 0.3 \times 0.3 \times 0.7 \times 0.9 \times 0.9 = 0.010206$ [SSSFFF] | A1 | Either correct. Accept unsimplified. |
| [Total $=$] $0.0160[38]$ | A1 | |
---6 Sajid is practising for a long jump competition. He counts any jump that is longer than 6 m as a success. On any day, the probability that he has a success with his first jump is 0.2 . For any subsequent jump, the probability of a success is 0.3 if the previous jump was a success and 0.1 otherwise. Sajid makes three jumps.
\begin{enumerate}[label=(\alph*)]
\item Draw a tree diagram to illustrate this information, showing all the probabilities.
\item Find the probability that Sajid has exactly one success given that he has at least one success.\\
On another day, Sajid makes six jumps.
\item Find the probability that only his first three jumps are successes or only his last three jumps are successes.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q6 [10]}}