Edexcel A-Level Mathematics A2 S2.1 The Binomial and Poisson Distributions Questions

Practise choosing binomial or Poisson models, calculating probabilities, using approximations, and interpreting mean and variance in S2 contexts.

Exam points

choose binomial or Poisson models from wording about trials, complaints or rare events
calculate exact, cumulative and complementary probabilities for counts
use E(X) and Var(X) to set parameters or comment on whether a Poisson model fits data

Covered syllabus points

S2.1.1 - Binomial and Poisson distributions S2.1.2 - Mean and variance of binomial and Poisson distributions S2.1.3 - The

[Maximum number: 5]

The length of pregnancy for a randomly selected pregnant sheep is D days where

D \sim \mathrm{~N}\left(112.4, \sigma^{2}\right)

Given that 5% of pregnant sheep have a length of pregnancy of less than 108 days,

(a)

Qiang selects 25 pregnant sheep at random from a large flock.

Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days.

[ 2 ]

$J\sim\mathrm{B}(25,0.05)$ $[\mathrm{P}(J\geq4)=]1-\mathrm{P}(J\leq3)=1-0.9659$ =0.0341 (calc $0.034090\ldots$ )

awrt 0.0341
(2)

(b)

Charlie takes 200 random samples of 25 pregnant sheep.

Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.

[ 3 ]

$T\sim \operatorname{Po}(200\times0.0341)=\operatorname{Po}(6.82)$ $\mathrm{P}(T>2)=1-\mathrm{P}(T\le 1)$ $=1-(\mathrm{e}^{-6.82}+\mathrm{e}^{-6.82}\times6.82)$ $=0.99146\ldots\quad(\text{calc }0.99144\ldots)$

[Maximum number: 8]

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!}=0.1281\ldots

M1 A1

(ii)

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

[ 2 ]

\mathrm{P}(4<S<10)=\mathrm{P}(S\le9)-\mathrm{P}(S\le4)=0.9829-0.5321=0.4508

M1 A1

(c)

Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected 20 m^2 of the peat bog is 0.012

A random sample of 200 non-overlapping 20 m^2 areas of the peat bog is taken.

Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids.

[ 3 ]

$V\sim \operatorname{Po}(200\times0.012)=\operatorname{Po}(2.4)$ $P(V=0)+P(V=1)=e^{-2.4}(1+2.4)$ $=0.30844\ldots$
Marking guidance:
M1 for writing or using $\operatorname{Po}(200\times0.012)$ .
dM1 for using P(V=0)+P(V=1), $e^{-\lambda}(1+\lambda)$ , or $P(V\le1)$ .
A1 for awrt 0.308.

M1 dM1 A1

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)