Poisson Approximation to Binomial (How to proceed)

Question

A charity issues a large number of certificates each costing $£10$ and each being repayable one year after issue. Of these certificates, $1$% are randomly selected to receive a prize of £10 such that they are repaid as $£20$. The remaining $99$% are repaid at their face value of $£10$.

Consider a person who purchases $200$ of these certificates.

Use a Poisson approximation to this binomial distribution to approximate the probability that this person is repaid more than $£2,040$.

My attempt,

I know that $N$~$Bin(200,0,01)$ can be approximated to $Poi(2)$

So I've to calculate $P(S>2040)$, but I don't know how to proceed.

Hope someone can explain it to me. Thanks in advance.

Answer 1

First, the Poisson approximation states $Bin(n,p)\simeq Poi(np)$ so we can use $Poi(2)$.

Now, The basic repay is £10, each prize is additional £10 so we'd like to inquire the probability of $k>4$ prizes, this can be found using $P(k>4)=1-P(k\leq4)$.

$$P(k\leq4)=P(k=0)+P(k=1)+P(k=2)+P(k=3)+P(k=4)=e^{-2}(\frac{2^0}{0!}+\frac{2^1}{1!}+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!})=e^{-2}(\frac{1}{1}+\frac{2}{1}+\frac{4}{2}+\frac{8}{6}+\frac{16}{24})=0.135335\cdot7=0.947347\Rightarrow P(k>4)=1-P(k\leq4)=0.052653$$

So the probability of gaining more than £2040 is 0.052653. Two thing you should note:

1. The Poisson approximation lets you work with the number of occurred events regardless of their order of appearance within the sequence of $n$ events.

2. This approximation assumes $n\rightarrow\infty,p\rightarrow0$. While everything holds in this question (in the manner of 'we did not get anything odd', as is the usual case with textbook questions), I'd say that for $n=200$ this is a quite risky case of using this approximation.