| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Mean/expectation of geometric distribution |
| Difficulty | Moderate -0.8 This is a straightforward application of standard geometric distribution formulas. Part (a) requires only recall of the mean formula (1/p = 6). Parts (b) and (c) involve routine probability calculations using P(X=r) = (1-p)^(r-1)p or cumulative probabilities, with no conceptual challenges or problem-solving required beyond direct formula application. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6\) | B1 | WWW |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(\frac{5}{6}\right)^3\frac{1}{6}+\left(\frac{5}{6}\right)^4\frac{1}{6}+\left(\frac{5}{6}\right)^5\frac{1}{6}+\left(\frac{5}{6}\right)^6\frac{1}{6}\) | M1 | \(p^3(1-p)+p^4(1-p)+p^5(1-p)+p^6(1-p)\), \(0 < p < 1\) |
| \(0.300\) \((0.2996\ldots)\) | A1 | At least 3 s.f. Award at most accurate value |
| Alternative: \(\left(\frac{5}{6}\right)^3-\left(\frac{5}{6}\right)^7\) | M1 | \(p^3-p^7\), \(0 < p < 1\) |
| \(0.300\) \((0.2996\ldots)\) | A1 | At least 3 s.f. Award at most accurate value |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1-\left(\frac{5}{6}\right)^9\) | M1 | \(1-p^n\), \(0 < p < 1\), \(n = 9, 10\) |
| \(0.806\) | A1 | |
| Alternative: \(\frac{1}{6}+\frac{1}{6}\left(\frac{5}{6}\right)+\frac{1}{6}\left(\frac{5}{6}\right)^2+\cdots+\frac{1}{6}\left(\frac{5}{6}\right)^8\) | M1 | \(p+p(1-p)+p(1-p)^2+\cdots+p(1-p)^8\) \((+p(1-p)^9)\), \(0 < p < 1\); as per answer for minimum terms shown |
| \(0.806\) | A1 | |
| Total: 2 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6$ | B1 | WWW |
| | **Total: 1** | |
---
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(\frac{5}{6}\right)^3\frac{1}{6}+\left(\frac{5}{6}\right)^4\frac{1}{6}+\left(\frac{5}{6}\right)^5\frac{1}{6}+\left(\frac{5}{6}\right)^6\frac{1}{6}$ | M1 | $p^3(1-p)+p^4(1-p)+p^5(1-p)+p^6(1-p)$, $0 < p < 1$ |
| $0.300$ $(0.2996\ldots)$ | A1 | At least 3 s.f. Award at most accurate value |
| **Alternative:** $\left(\frac{5}{6}\right)^3-\left(\frac{5}{6}\right)^7$ | M1 | $p^3-p^7$, $0 < p < 1$ |
| $0.300$ $(0.2996\ldots)$ | A1 | At least 3 s.f. Award at most accurate value |
| | **Total: 2** | |
---
**Part (c):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1-\left(\frac{5}{6}\right)^9$ | M1 | $1-p^n$, $0 < p < 1$, $n = 9, 10$ |
| $0.806$ | A1 | |
| **Alternative:** $\frac{1}{6}+\frac{1}{6}\left(\frac{5}{6}\right)+\frac{1}{6}\left(\frac{5}{6}\right)^2+\cdots+\frac{1}{6}\left(\frac{5}{6}\right)^8$ | M1 | $p+p(1-p)+p(1-p)^2+\cdots+p(1-p)^8$ $(+p(1-p)^9)$, $0 < p < 1$; as per answer for minimum terms shown |
| $0.806$ | A1 | |
| | **Total: 2** | |1 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable $X$.
\begin{enumerate}[label=(\alph*)]
\item Write down the mean of $X$.
\item Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw.
\item Find the probability that a 5 is first obtained in fewer than 10 throws.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q1 [5]}}