Edexcel IAL Mathematics S2.1.3 Poisson approximation to the binomial

Practise replacing suitable binomial models with Poisson approximations and using the new parameter to estimate rare-event probabilities.

Exam points

convert B(n,p) to Po(np) when n is large and p is small, then calculate a tail probability

[Maximum number: 3]

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

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.

$V\sim \operatorname{Po}(200\times0.012)=\operatorname{Po}(2.4)$ $P(V=0)+P(V=1)=e^{-2.4}(1+2.4)$ $=0.30844\ldots$

M1 dM1 A1

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.

All question bank results loaded

S2.1.3 - The question 1