| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Compound event with two dice/coins |
| Difficulty | Moderate -0.8 This is a straightforward geometric distribution question with standard calculations. Part (a) is basic enumeration of outcomes, parts (b)-(d) apply standard geometric distribution formulas (mean = 1/p, P(X=k) = (1-p)^(k-1)p, cumulative probability). All steps are routine with no problem-solving insight required beyond recognizing the geometric distribution setup. |
| Spec | 2.03a Mutually exclusive and independent events5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Prob of 4 (from 1,3, 3,1 or 2,2) \(= \frac{3}{36} = \frac{1}{12}\) | B1 | AG |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= \frac{1}{\frac{1}{12}} = 12\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left(\frac{11}{12}\right)^5 \times \frac{1}{12} = 0.0539\) or \(\frac{161051}{2985984}\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1 - \left(\frac{11}{12}\right)^7\) | M1 | |
| \(0.456\) or \(\frac{16344637}{35831808}\) | A1 | |
| Total: 2 |
## Question 1:
**Part (a)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Prob of 4 (from 1,3, 3,1 or 2,2) $= \frac{3}{36} = \frac{1}{12}$ | B1 | AG |
| **Total: 1** | | |
---
**Part (b)**
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= \frac{1}{\frac{1}{12}} = 12$ | B1 | |
| **Total: 1** | | |
---
**Part (c)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left(\frac{11}{12}\right)^5 \times \frac{1}{12} = 0.0539$ or $\frac{161051}{2985984}$ | B1 | |
| **Total: 1** | | |
---
**Part (d)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 - \left(\frac{11}{12}\right)^7$ | M1 | |
| $0.456$ or $\frac{16344637}{35831808}$ | A1 | |
| **Total: 2** | | |1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.
\begin{enumerate}[label=(\alph*)]
\item Show that the probability that the score is 4 is $\frac { 1 } { 12 }$.\\
The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable $X$.
\item Find the mean of $X$.
\item Find the probability that a score of 4 is first obtained on the 6th throw.
\item Find $\mathrm { P } ( X < 8 )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q1 [5]}}