| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Conditional with three or more stages |
| Difficulty | Standard +0.3 This is a straightforward multi-stage probability tree problem with clear probabilities given. Part (a) is routine calculation, part (b) requires combining branches, and part (c) involves basic conditional probability. All steps follow standard S1 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.6 + 0.4 \times 0.3 = 0.72\) or \(1 - 0.4 \times 0.7 = 0.72\) | B1 | Clear identified calculation AG |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.72 \times (0.4 + 0.6 \times 0.2)\) | M1 | \(0.72 \times u\), \(0 < u < 1\) |
| M1 | \(v \times (0.4 + 0.6 \times 0.2)\), or \(v \times (1-0.6\times 0.8)\), \(0 < v \leqslant 1\); SC B1 for \(0.72\times(0.4+0.12)\) or \(0.72\times(1-0.48)\) | |
| \(0.3744\) | A1 | WWW. Condone 0.374. SC B1 for 0.3744 only |
| 3 | ||
| Alternative: \([p(P1P2)+p(F1P1P2)+p(P1F2P2)+p(F1P1F2P2)]\) \(= 0.6\times0.4+0.4\times0.3\times0.4+0.6\times0.6\times0.2+0.4\times0.3\times0.6\times0.2\) | M1 | Any two terms unsimplified and correct |
| M1 | Summing 4 appropriate scenarios; SC B1 for \(0.24+0.048+0.072+0.0144\) | |
| \(0.3744\) | A1 | WWW. Condone 0.374. SC B1 for 0.3744 only |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{fails first or second level} \mid \text{finishes game}) = \frac{P(\text{fails first or second level} \cap \text{finishes game})}{\textit{their } \mathbf{(b)}}\) | M1 | Either \(0.6\times0.6\times0.2\) or \(0.4\times0.3\times0.4\) seen; condone 0.072 or 0.048 if seen in (b) |
| Numerator \(= P(S\ SF)+P(FS\ S) = 0.6\times0.6\times0.2+0.4\times0.3\times0.4 = 0.072+0.048=0.12\) | A1 | Both correct, accept unsimplified. No additional terms |
| Required probability \(= \frac{0.12}{\textit{their }\mathbf{(b)}}\) | M1 | *Their* sum of two 3-term probabilities as numerator over *their* (b) or correct |
| \(0.321\) or \(\frac{25}{78}\) | A1 | \(0.3205 < p \leqslant 0.321\) |
| 4 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.6 + 0.4 \times 0.3 = 0.72$ or $1 - 0.4 \times 0.7 = 0.72$ | B1 | Clear identified calculation AG |
| | **1** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.72 \times (0.4 + 0.6 \times 0.2)$ | M1 | $0.72 \times u$, $0 < u < 1$ |
| | M1 | $v \times (0.4 + 0.6 \times 0.2)$, or $v \times (1-0.6\times 0.8)$, $0 < v \leqslant 1$; **SC B1** for $0.72\times(0.4+0.12)$ or $0.72\times(1-0.48)$ |
| $0.3744$ | A1 | WWW. Condone 0.374. **SC B1** for 0.3744 only |
| | **3** | |
| **Alternative:** $[p(P1P2)+p(F1P1P2)+p(P1F2P2)+p(F1P1F2P2)]$ $= 0.6\times0.4+0.4\times0.3\times0.4+0.6\times0.6\times0.2+0.4\times0.3\times0.6\times0.2$ | M1 | Any two terms unsimplified and correct |
| | M1 | Summing 4 appropriate scenarios; **SC B1** for $0.24+0.048+0.072+0.0144$ |
| $0.3744$ | A1 | WWW. Condone 0.374. **SC B1** for 0.3744 only |
| | **3** | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{fails first or second level} \mid \text{finishes game}) = \frac{P(\text{fails first or second level} \cap \text{finishes game})}{\textit{their } \mathbf{(b)}}$ | M1 | Either $0.6\times0.6\times0.2$ or $0.4\times0.3\times0.4$ seen; condone 0.072 or 0.048 if seen in **(b)** |
| Numerator $= P(S\ SF)+P(FS\ S) = 0.6\times0.6\times0.2+0.4\times0.3\times0.4 = 0.072+0.048=0.12$ | A1 | Both correct, accept unsimplified. No additional terms |
| Required probability $= \frac{0.12}{\textit{their }\mathbf{(b)}}$ | M1 | *Their* sum of two 3-term probabilities as numerator over *their* **(b)** or correct |
| $0.321$ or $\frac{25}{78}$ | A1 | $0.3205 < p \leqslant 0.321$ |
| | **4** | |6 Janice is playing a computer game. She has to complete level 1 and level 2 to finish the game. She is allowed at most two attempts at any level.
\begin{itemize}
\item For level 1 , the probability that Janice completes it at the first attempt is 0.6 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.3 .
\item If Janice completes level 1, she immediately moves on to level 2.
\item For level 2, the probability that Janice completes it at the first attempt is 0.4 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.2 .
\begin{enumerate}[label=(\alph*)]
\item Show that the probability that Janice moves on to level 2 is 0.72 .
\item Find the probability that Janice finishes the game.
\item Find the probability that Janice fails exactly one attempt, given that she finishes the game.\\
\end{itemize}
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q6 [8]}}