Question

HMRC report that 5% of taxpayers filling out the short form make serious mistakes. In a survey, 50 of these forms are chosen at random.

Find the probability that at most two forms have serious mistakes. Perform the calculation:

1. using the Binomial distribution,
2. using the Poisson approximation to the binomial distribution.

Discuss whether you consider the approximation is appropriate.

Answer 1

a)

Let X denote the number of forms that contain mistakes. Then $X\sim Bin(n=50,p=0.05)$

$P(X=x)=\binom{50}{x}(0.05)^x(1-0.05)^{50-x},x=0,1,2,....,50$

Required probability = $P(X\leq 2)=P(X=0)+P(X=1)+P(X=2)$

$=\binom{50}{0}(0.05)^0(1-0.05)^{50-0}+\binom{50}{1}(0.05)^1(1-0.05)^{50-1}+\binom{50}{2}(0.05)^2(1-0.05)^{50-2}$

$=0.5405$

b)

Using Poisson approximation, $X\sim Poi(\lambda=np=50(0.05)=2.5)$

$P(X=x)=\frac{2.5^2e^{-2.5}}{x!},x=0,1,2,3,....$

Required probability = $P(X\leq 2)=P(X=0)+P(X=1)+P(X=2)$

$=\frac{2.5^0e^{-2.5}}{0!}+\frac{2.5^1e^{-2.5}}{1!}+\frac{2.5^2e^{-2.5}}{2!}=0.5438$

If 'n' is greater than 20 and 'np' is less than 5, then Poisson approximation is a good approximation.

Here, n=50 and np=2.5. Hence, the approximation is appropriate.