CAIE S2 2020 March — Question 1 3 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2020
SessionMarch
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPoisson distribution
TypeStandard Poisson approximation to binomial
DifficultyModerate -0.3 This is a straightforward application of Poisson approximation to binomial with clearly stated parameters (n=12500, p=1/30000, giving λ=0.4167). The calculation requires only finding P(X≥2)=1-P(X≤1) using standard Poisson probability formula, which is routine for S2 level with no conceptual challenges or multi-step reasoning.
Spec5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!
1 The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30000 letters, and these errors occur randomly. A randomly chosen booklet from this publisher contains 12500 letters. Use a suitable approximating distribution to find the probability that this booklet contains at least 2 errors.
Question 1:
AnswerMarks Guidance
AnswerMark Guidance
\((\lambda =) \frac{5}{12} = 0.417\) or betterB1
\(1 - e^{-\frac{5}{12}}(1 + \frac{5}{12})\)M1 \(1 - P(X = 0 \text{ or } 1)\), by Poisson, using any \(\lambda\), allow \(1 - P(X = 0 \text{ or } 1 \text{ or } 2)\) for M1
\(= 0.0661\) or \(0.0662\) (3 sf)A1 Final answer. SC use of Binomial (from \(0.06607...\)) B1 only
3 
## Question 1:

| Answer | Mark | Guidance |
|--------|------|----------|
| $(\lambda =) \frac{5}{12} = 0.417$ or better | **B1** | |
| $1 - e^{-\frac{5}{12}}(1 + \frac{5}{12})$ | **M1** | $1 - P(X = 0 \text{ or } 1)$, by Poisson, using any $\lambda$, allow $1 - P(X = 0 \text{ or } 1 \text{ or } 2)$ for M1 |
| $= 0.0661$ or $0.0662$ (3 sf) | **A1** | Final answer. **SC** use of Binomial (from $0.06607...$) B1 only |
| | **3** | |

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1 The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30000 letters, and these errors occur randomly. A randomly chosen booklet from this publisher contains 12500 letters.

Use a suitable approximating distribution to find the probability that this booklet contains at least 2 errors.\\

\hfill \mbox{\textit{CAIE S2 2020 Q1 [3]}}
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