| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Geometric then binomial separate scenarios |
| Difficulty | Moderate -0.8 Part (a) is a straightforward geometric distribution calculation requiring P(X < 7) = 1 - (4/5)^6. Part (b) is a standard binomial probability P(5 ≤ X ≤ 7) requiring summation of three terms. Both parts involve direct application of standard formulas with no conceptual challenges or problem-solving insight required, making this easier than average. |
| 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 | Mark | Guidance |
| \([P(5) = 0.2]\); \([P(X < 7) =] 1 - 0.8^6\) | M1 | \(1 - 0.8^n\), \(n = 6, 7\) |
| \(= 0.738, \frac{11529}{15625}\) | A1 | 0.737856 to at least 3SF |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([P(X<7)=]\ 0.2 + 0.2\times0.8 + 0.2\times0.8^2 + 0.2\times0.8^3 + 0.2\times0.8^4 + 0.2\times0.8^5\) | M1 | \(0.2 + 0.2\times0.8 + 0.2\times0.8^2 + 0.2\times0.8^3 + 0.2\times0.8^4 + 0.2\times0.8^5\ (+0.2\times0.8^6)\) |
| \(= 0.738, \frac{11529}{15625}\) | A1 | 0.737856 to at least 3SF |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([P(5,6,7)=]\ {}^{10}C_5(0.2)^5(0.8)^5 + {}^{10}C_6(0.2)^6(0.8)^4 + {}^{10}C_7(0.2)^7(0.8)^3\) | M1 | One term: \({}^{10}C_x(p)^x(1-p)^{10-x}\), \(0 < p < 1\), \(x \neq 0, 10\) |
| \([0.02642 + 5.505\times10^{-3} + 7.864\times10^{-4}]\) | A1 | Correct expression, accept unsimplified, no terms omitted leading to final answer |
| \(= 0.0327\) | B1 | awrt |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([P(X<8) - P(X\leqslant4) = 1 - P(X\geqslant8) - P(X\leqslant4) =]\) \(1 - \{{}^{10}C_8(0.2)^8(0.8)^2 + {}^{10}C_9(0.2)^90.8 + (0.2)^{10}\}\) \(- \{(0.8)^{10} + {}^{10}C_1(0.2)(0.8)^9 + {}^{10}C_2(0.2)^2(0.8)^8 + {}^{10}C_3(0.2)^3(0.8)^7 + {}^{10}C_4(0.2)^4(0.8)^6\}\) | M1 | One term: \({}^{10}C_x(p)^x(1-p)^{10-x}\), \(0 < p < 1\), \(x \neq 0, 10\) |
| \([1 - \{7.373\times10^{-5} + 4.096\times10^{-6} + 1.024\times10^{-7}\}] - \{0.1074 + 0.2684 + 0.3020 + 0.2013 + 0.08808\}\) | A1 | Correct expression, accept unsimplified, no terms omitted leading to final answer |
| \(= 0.0327\) | B1 | awrt |
| 3 |
## Question 2(a):
**Method 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(5) = 0.2]$; $[P(X < 7) =] 1 - 0.8^6$ | M1 | $1 - 0.8^n$, $n = 6, 7$ |
| $= 0.738, \frac{11529}{15625}$ | A1 | 0.737856 to at least 3SF |
**Method 2:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(X<7)=]\ 0.2 + 0.2\times0.8 + 0.2\times0.8^2 + 0.2\times0.8^3 + 0.2\times0.8^4 + 0.2\times0.8^5$ | M1 | $0.2 + 0.2\times0.8 + 0.2\times0.8^2 + 0.2\times0.8^3 + 0.2\times0.8^4 + 0.2\times0.8^5\ (+0.2\times0.8^6)$ |
| $= 0.738, \frac{11529}{15625}$ | A1 | 0.737856 to at least 3SF |
| | **2** | |
---
## Question 2(b):
**Method 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(5,6,7)=]\ {}^{10}C_5(0.2)^5(0.8)^5 + {}^{10}C_6(0.2)^6(0.8)^4 + {}^{10}C_7(0.2)^7(0.8)^3$ | M1 | One term: ${}^{10}C_x(p)^x(1-p)^{10-x}$, $0 < p < 1$, $x \neq 0, 10$ |
| $[0.02642 + 5.505\times10^{-3} + 7.864\times10^{-4}]$ | A1 | Correct expression, accept unsimplified, no terms omitted leading to final answer |
| $= 0.0327$ | B1 | awrt |
**Method 2:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(X<8) - P(X\leqslant4) = 1 - P(X\geqslant8) - P(X\leqslant4) =]$ $1 - \{{}^{10}C_8(0.2)^8(0.8)^2 + {}^{10}C_9(0.2)^90.8 + (0.2)^{10}\}$ $- \{(0.8)^{10} + {}^{10}C_1(0.2)(0.8)^9 + {}^{10}C_2(0.2)^2(0.8)^8 + {}^{10}C_3(0.2)^3(0.8)^7 + {}^{10}C_4(0.2)^4(0.8)^6\}$ | M1 | One term: ${}^{10}C_x(p)^x(1-p)^{10-x}$, $0 < p < 1$, $x \neq 0, 10$ |
| $[1 - \{7.373\times10^{-5} + 4.096\times10^{-6} + 1.024\times10^{-7}\}] - \{0.1074 + 0.2684 + 0.3020 + 0.2013 + 0.08808\}$ | A1 | Correct expression, accept unsimplified, no terms omitted leading to final answer |
| $= 0.0327$ | B1 | awrt |
| | **3** | |
---2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that it takes fewer than 7 spins for George to obtain a 5 .\\
George spins the spinner 10 times.
\item Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q2 [5]}}