| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Compound event with two dice/coins |
| Difficulty | Moderate -0.3 This is a straightforward geometric distribution application with clearly defined success probability (1/4 for two tails). Part (a) requires direct formula substitution P(X=7)=(3/4)^6(1/4), and part (b) needs P(X>9)=(3/4)^9. Both are standard textbook exercises requiring only recall of geometric distribution formulas with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(\frac{3}{4}\right)^6 \frac{1}{4}\) | M1 | \((1-p)^6 p\), \(0 < p < 1\) |
| \(0.0445\), \(\frac{729}{16384}\) | A1 |
Total: 2 marks
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(\frac{3}{4}\right)^9\) | M1 | \(\left(\frac{3}{4}\right)^n\) or \(p^n\), \(0 < p < 1\), \(n = 8, 9, 10\) |
| \(0.0751\), \(\frac{19683}{262144}\) | A1 |
Total: 2 marks
**Question 1:**
**Part 1(a):**
$\left(\frac{3}{4}\right)^6 \frac{1}{4}$ | M1 | $(1-p)^6 p$, $0 < p < 1$
$0.0445$, $\frac{729}{16384}$ | A1 |
Total: **2 marks**
---
**Part 1(b):**
$\left(\frac{3}{4}\right)^9$ | M1 | $\left(\frac{3}{4}\right)^n$ or $p^n$, $0 < p < 1$, $n = 8, 9, 10$
$0.0751$, $\frac{19683}{262144}$ | A1 |
Total: **2 marks**
---1 Two fair coins are thrown at the same time. The random variable $X$ is the number of throws of the two coins required to obtain two tails at the same time.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that two tails are obtained for the first time on the 7th throw.
\item Find the probability that it takes more than 9 throws to obtain two tails for the first time.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q1 [4]}}