| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | First success on specific trial |
| Difficulty | Moderate -0.3 Part (a) requires systematic enumeration of outcomes (1+1+2, 1+2+1, 2+1+1) giving probability 3/216. Part (b) applies the geometric distribution formula directly: (1-p)^4 × p where p=1/216 for score 18. This is a straightforward application of standard probability concepts with minimal computational complexity, making it slightly easier than average for A-level. |
| Spec | 2.04a Discrete probability distributions5.02f Geometric distribution: conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Possible cases: 1 1 2, 1 2 1, 2 1 1; Probability \(= \left(\frac{1}{6}\right)^3 \times 3\) | M1 | \(\left(\frac{1}{6}\right)^3 \times k\), where \(k\) is an integer |
| M1 | Multiply a probability by 3, not \(+\), \(-\) or \(\div\) | |
| \(\frac{1}{72}\) | A1 | Accept \(\frac{3}{216}\) or \(0.013\overline{8}\) or 0.0139 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(18) = \left(\frac{1}{6}\right)^3 \left[= \frac{1}{216}\right]\) | B1 | |
| \(P(\text{18 on 5th throw}) = \left(\frac{215}{216}\right)^4 \times \frac{1}{216}\) | M1 | \((1-p)^4 p,\ 0 < their\ p < 1\) |
| 0.00454 | A1 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Possible cases: 1 1 2, 1 2 1, 2 1 1; Probability $= \left(\frac{1}{6}\right)^3 \times 3$ | M1 | $\left(\frac{1}{6}\right)^3 \times k$, where $k$ is an integer |
| | M1 | Multiply a probability by 3, not $+$, $-$ or $\div$ |
| $\frac{1}{72}$ | A1 | Accept $\frac{3}{216}$ or $0.013\overline{8}$ or 0.0139 |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(18) = \left(\frac{1}{6}\right)^3 \left[= \frac{1}{216}\right]$ | B1 | |
| $P(\text{18 on 5th throw}) = \left(\frac{215}{216}\right)^4 \times \frac{1}{216}$ | M1 | $(1-p)^4 p,\ 0 < their\ p < 1$ |
| 0.00454 | A1 | |
---4 Three fair six-sided dice, each with faces marked $1,2,3,4,5,6$, are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the score is 4 on a single throw of the three dice.
\item Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q4 [6]}}