Poisson approximation to the binomialEdexcel International A Level Further Maths: Flashcards

What these 12 flashcards ask

When can B(n,p) be approximated by a Poisson distribution?
State the approximating distribution for X\sim B(n,p).
X\sim B(100,0.03). Give the approximating distribution.
Give typical numerical guides for the approximation.
Why are the mean and variance of B(n,p) close when p is small?
What is the variance of Po(\lambda)?
X\sim B(200,0.01). Write P(X\le1) using the approximation.
Is B(10,0.4) well approximated by a Poisson distribution?
Can a Poisson variable take values larger than n?
How do you comment on the accuracy of the approximation?
X\approxPo(4). Write P(X\ge7) in terms of P(X\le r).
Why use the approximation at all?

Exam questions on Poisson approximation to the binomial

A factory produces light bulbs. Each bulb is independently defective with probability 0.02. A box contains 200 bulbs, and $X$ is the number of defective bulbs in a box, so $X\sim B(200,0.02)$ .
Use a Poisson approximation to find the probability that a box contains more than 5 defective bulbs.2 marks
An airline finds that each passenger booked on a flight independently fails to turn up with probability 0.03. A flight has 120 passengers booked, and $X$ is the number who fail to turn up, so $X\sim B(120,0.03)$ .
Use the Poisson approximation to find the probability that at most 2 passengers fail to turn up.2 marks
A rare genetic condition affects 1 in 500 newborn babies, independently of one another. In one year a region has 1000 births, and $X$ is the number of those babies born with the condition.
State the exact distribution of $X$ , name a suitable approximating distribution, and give a reason why the approximation is suitable.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).