Question 1 - A-Level Maths

CAIE S2 2024 November — Question 1 3 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2024
Session	November
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Approximating the Binomial to the Poisson distribution
Type	Calculate single probability using Poisson approximation
Difficulty	Moderate -0.5 This is a straightforward application of the standard Poisson approximation to the binomial. Students need to calculate λ = np = 4.5, recognize they need P(X ≥ 4) = 1 - P(X ≤ 3), then sum Poisson probabilities. While it requires careful calculation, it's a routine textbook exercise with no conceptual challenges beyond knowing when and how to apply the approximation.
Spec	2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial 5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\lambda = 4.5\)	B1
\(1 - e^{-4.5}(1 + 4.5 + \frac{4.5^2}{2} + \frac{4.5^3}{3!}) = 1 - e^{-4.5}(1 + 4.5 + 10.125 + 15.1875)\) \(= 1-(0.011109 + 0.049999 + 0.11248 + 0.16872)\)	M1	Expression must be seen or implied by correct figures. Any \(\lambda\). Allow one end error. Accept fully correct \(\Sigma\) notation.
\(0.658\) (3 sf)	A1	SC unsupported 0.658 scores B1 B1.
Total: 3

**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 4.5$ | B1 | |
| $1 - e^{-4.5}(1 + 4.5 + \frac{4.5^2}{2} + \frac{4.5^3}{3!}) = 1 - e^{-4.5}(1 + 4.5 + 10.125 + 15.1875)$ $= 1-(0.011109 + 0.049999 + 0.11248 + 0.16872)$ | M1 | Expression must be seen or implied by correct figures. Any $\lambda$. Allow one end error. Accept fully correct $\Sigma$ notation. |
| $0.658$ (3 sf) | A1 | SC unsupported 0.658 scores **B1 B1**. |
| **Total: 3** | | |

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1 A random variable $X$ has the distribution $\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)$.\\
Use a Poisson distribution to calculate an estimate of $\mathrm { P } ( X \geqslant 4 )$.\\

\includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}

\hfill \mbox{\textit{CAIE S2 2024 Q1 [3]}}

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