| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Geometric then binomial separate scenarios |
| Difficulty | Moderate -0.8 This question tests standard applications of geometric and binomial distributions with straightforward probability calculations. Part (a) requires P(X > 6) = (0.75)^6 using the geometric distribution, and part (b) requires P(X ≥ 3) = 1 - P(X ≤ 2) using the binomial distribution. Both are routine textbook exercises requiring only formula recall and basic calculator work, with no problem-solving or conceptual insight needed. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(X > 6) = 0.75^6\) | M1 | \(p^n\), \(n = 6, 7\), \(0 < p < 1\) |
| \(0.178, \frac{729}{4096}\) | A1 | \(0.17797\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1 - P(0,1,2) = 1-(0.75^{10} + {}^{10}C_1\ 0.25^1\ 0.75^9 + {}^{10}C_2\ 0.25^2\ 0.75^8)\) | M1 | Binomial term of form \({}^{10}C_x\ p^x(1-p)^{10-x}\), \(0 < p < 1\), any \(p\), \(x \neq 0, 10\) |
| \(1-(0.0563135 + 0.1877117 + 0.2815676)\) | A1 | Correct unsimplified expression |
| \(0.474\) | A1 | \(0.474 \leqslant p \leqslant 0.4744\) |
## Question 3:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(X > 6) = 0.75^6$ | M1 | $p^n$, $n = 6, 7$, $0 < p < 1$ |
| $0.178, \frac{729}{4096}$ | A1 | $0.17797\ldots$ |
**Total: 2 marks**
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1 - P(0,1,2) = 1-(0.75^{10} + {}^{10}C_1\ 0.25^1\ 0.75^9 + {}^{10}C_2\ 0.25^2\ 0.75^8)$ | M1 | Binomial term of form ${}^{10}C_x\ p^x(1-p)^{10-x}$, $0 < p < 1$, any $p$, $x \neq 0, 10$ |
| $1-(0.0563135 + 0.1877117 + 0.2815676)$ | A1 | Correct unsimplified expression |
| $0.474$ | A1 | $0.474 \leqslant p \leqslant 0.4744$ |
**Total: 3 marks**
---3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .
\begin{enumerate}[label=(\alph*)]
\item Find the probability that Kayla takes more than 6 throws to achieve a success.
\item Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q3 [5]}}