Question 1 - A-Level Maths

CAIE S2 Specimen — Question 1 4 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Session	Specimen
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Sum of Poisson processes
Type	Basic sum of two Poissons
Difficulty	Standard +0.3 This requires knowing that independent Poisson distributions sum to another Poisson distribution, adjusting rates for 6 months (0.6 + 1.15 = 1.75), then calculating P(X=2) + P(X=3) using standard Poisson formula. Straightforward application of theory with routine calculations, slightly above average due to the multi-step setup but no novel insight required.
Spec	5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.02n Sum of Poisson variables: is Poisson

1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\lambda = (1.2 + 2.3) \div 2\)	M1	Attempt combined mean, allow \(1.2 + 2.3\)
\(= 1.75\)	A1	Correct mean
\(e^{-1.75}\left(\frac{1.75^2}{2} + \frac{1.75^3}{3!}\right)\)	M1	Allow incorrect mean
\(= 0.421\) (3 sf)	A1	Allow end errors (1 and/or 4)
Total	4

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = (1.2 + 2.3) \div 2$ | M1 | Attempt combined mean, allow $1.2 + 2.3$ |
| $= 1.75$ | A1 | Correct mean |
| $e^{-1.75}\left(\frac{1.75^2}{2} + \frac{1.75^3}{3!}\right)$ | M1 | Allow incorrect mean |
| $= 0.421$ (3 sf) | A1 | Allow end errors (1 and/or 4) |
| **Total** | **4** | |

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1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .\\

\hfill \mbox{\textit{CAIE S2  Q1 [4]}}

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