Pearson Edexcel IAL Mathematics S2.1.1 Binomial & Poisson distributions

Practise modelling count data with binomial and Poisson distributions, including cumulative probabilities, approximations and suitability comments.

Exam points

calculate exact and cumulative count probabilities using binomial or Poisson formulae
decide when a Poisson model is suitable by comparing the sample mean and variance
form a second-stage binomial model when repeated samples meet a stated condition

[Maximum number: 5]

In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per $\mathrm{m}^{2}$

(a)

Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per m^2 of the peat bog to follow a Poisson distribution.

[ 1 ]

(b)

Given that the number of Common Spotted-orchids in 1 m^2 of the peat bog can be modelled by a Poisson distribution,

find the probability that in a randomly selected 1 m^2 of the peat bog

[ 4 ]

(i)

there are exactly 6 Common Spotted-orchids,

[ 2 ]

$S \sim \operatorname{Po}(4.5)$ $\mathrm{P}(S=6)=\frac{\mathrm{e}^{-4.5} 4.5^{6}}{6!}$ or $\mathrm{P}(S \leq 6)-\mathrm{P}(S \leq 5)$

M1

$=0.1281 \ldots$

A1

(ii)

there are fewer than 10 but more than 4 Common Spotted-orchids.

[ 2 ]

$\mathrm{P}(4<S<10)=\mathrm{P}(S \leq 9)-\mathrm{P}(S \leq 4)$ or 0.9829-0.5321

M1

=0.4508

A1

(4)

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S2.1.1 - Binomial and Poisson distributions question 1

Question (a)

Question (b)

Question (i)

Question (ii)