| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Negative Binomial Distribution |
| Type | First success on specific trial |
| Difficulty | Moderate -0.8 This is a straightforward application of geometric and negative binomial distribution formulas with no conceptual challenges. Part (a) uses the basic geometric distribution formula P(X=k) = (1-p)^(k-1) × p, part (b) requires summing a geometric series or using complement rule, and part (c) applies the standard negative binomial formula. All are direct substitutions into standard formulas with minimal problem-solving required, making this easier than average. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([(0.7)^6 \times 0.3] = 0.0353\) | B1 | \(\frac{352947}{10000000}\) or \(0.03529\ldots\) to at least 3sf |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([P(X < 6) =] \ 1 - 0.7^5\) | M1 | \(1 - 0.7^d\), \(d = 5, 6\) |
| \(= 0.832\) | A1 | Accept \(0.83193\) to at least 3sf. If M0 scored, SC B1 for \(0.8319[3]\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([P(X < 6) =] \ 0.3 + (0.3)(0.7) + (0.3)(0.7)^2 + (0.3)(0.7)^3 + (0.3)(0.7)^4\) | (M1) | \(0.3 + (0.3)(0.7) + (0.3)(0.7)^2 + (0.3)(0.7)^3 + (0.3)(0.7)^4 \ [+ (0.3)(0.7)^{\ldots}]\) |
| \(= 0.832\) | (A1) | Accept \(0.83193\) to at least 3sf. If M0 scored, SC B1 for \(0.8319[3]\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0.7)^8 \times (0.3)^2 \times {}^9C_1\) or \((0.7)^8 \times (0.3) \times {}^9C_1 \times (0.3)\) | M1 | \((0.7)^8 \times (0.3)^2 \times k\), \(k\) a positive integer, 1 may be implied. No addition/subtraction/additional terms |
| \(= 0.0467\) | A1 | |
| Total: 2 |
## Question 1:
**Part (a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[(0.7)^6 \times 0.3] = 0.0353$ | B1 | $\frac{352947}{10000000}$ or $0.03529\ldots$ to at least 3sf |
| | **Total: 1** | |
**Part (b) — Method 1**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[P(X < 6) =] \ 1 - 0.7^5$ | M1 | $1 - 0.7^d$, $d = 5, 6$ |
| $= 0.832$ | A1 | Accept $0.83193$ to at least 3sf. If M0 scored, **SC B1** for $0.8319[3]$ |
| | **Total: 2** | |
**Part (b) — Method 2**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[P(X < 6) =] \ 0.3 + (0.3)(0.7) + (0.3)(0.7)^2 + (0.3)(0.7)^3 + (0.3)(0.7)^4$ | (M1) | $0.3 + (0.3)(0.7) + (0.3)(0.7)^2 + (0.3)(0.7)^3 + (0.3)(0.7)^4 \ [+ (0.3)(0.7)^{\ldots}]$ |
| $= 0.832$ | (A1) | Accept $0.83193$ to at least 3sf. If M0 scored, **SC B1** for $0.8319[3]$ |
| | **Total: 2** | |
**Part (c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.7)^8 \times (0.3)^2 \times {}^9C_1$ or $(0.7)^8 \times (0.3) \times {}^9C_1 \times (0.3)$ | M1 | $(0.7)^8 \times (0.3)^2 \times k$, $k$ a positive integer, 1 may be implied. No addition/subtraction/additional terms |
| $= 0.0467$ | A1 | |
| | **Total: 2** | |
---1 Rajesh applies once every year for a ticket to a music festival. The probability that he is successful in any particular year is 0.3 , independently of other years.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that Rajesh is successful for the first time on his 7th attempt.
\item Find the probability that Rajesh is successful for the first time before his 6th attempt.
\item Find the probability that Rajesh is successful for the second time on his 10th attempt.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q1 [5]}}