CAIE S1 2009 November — Question 1 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2009
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicBinomial Distribution
TypeBinomial with derived parameters
DifficultyModerate -0.3 This is a straightforward binomial probability calculation requiring students to first derive p=0.08 from the given mean (np=1.6, n=20), then calculate P(X>2)=1-P(X≤2). While it involves multiple steps and understanding the relationship between mean and parameters, it's a standard S1 question with routine application of binomial probability formulas.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities
1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(20p = 1.6\), \(p = 0.08\)M1 Equation relating \(20p\) to the mean
Correct \(p\)A1 \(p\) can be implied
\(P(X>2) = 1 - \{(0.92)^{20} + {}^{20}C_1(0.08)(0.92)^{19} + {}^{20}C_2(0.08)^2(0.92)^{18}\}\)M1 Bin expression involving \(p^x(1-p)^{20-x}\ {}^{20}C_x\) any \(p\)
\(= 1-(0.1887+0.3281+0.2711)\)M1 Subtracting 2 or 3 binomial probs from 1, one of which is \(P(0)\)
\(= 0.212\)A1 [5] Correct answer 
# Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $20p = 1.6$, $p = 0.08$ | M1 | Equation relating $20p$ to the mean |
| Correct $p$ | A1 | $p$ can be implied |
| $P(X>2) = 1 - \{(0.92)^{20} + {}^{20}C_1(0.08)(0.92)^{19} + {}^{20}C_2(0.08)^2(0.92)^{18}\}$ | M1 | Bin expression involving $p^x(1-p)^{20-x}\ {}^{20}C_x$ any $p$ |
| $= 1-(0.1887+0.3281+0.2711)$ | M1 | Subtracting 2 or 3 binomial probs from 1, one of which is $P(0)$ |
| $= 0.212$ | A1 **[5]** | Correct answer |

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1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.

\hfill \mbox{\textit{CAIE S1 2009 Q1 [5]}}
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