CAIE A-Level Mathematics 6.1.4 Poisson Approximation to the Binomial

Practise replacing a binomial model by Po(np) when trials are numerous and success is rare, justifying the choice and evaluating exact or cumulative probabilities.

Exam points

check that n is large and p is small, using the stated guidance n > 50 and np < 5
set the approximating Poisson mean to λ = np and state the complete Po(λ) model
translate the binomial event without continuity correction and evaluate its Poisson sum or tail

[Maximum number: 4]

It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.

(a)

Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.

[ 3 ]

$X\sim \operatorname{Po}\left(\frac{2}{3}\right)$
B1 Poisson with correct mean stated to at least 3 s.f. or implied in working.
$1-e^{-2/3}\left(1+\frac{2}{3}\right)$
M1 $1-\mathrm{P}(X=0\text{ or }1)$ ; allow one end error.
A1 0.144 (3 s.f.).
SC B1 for use of binomial or no working shown leading to correct final answer.

(b)

Justify the approximating distribution used in part (a).

The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.

[ 1 ]

$n>50 \quad\text{and}\quad np=\frac{2}{3}<5$
or
$n>50 \quad\text{and}\quad p=\frac{1}{300}<0.1$
B1 Accept p or np clearly stated in part (a). Do not accept only "n is large and p is small".

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6.1.4—Binomial distribution question 1

Question (a)

Question (b)