Answer to Question #72706 in Statistics and Probability for Danish

Question

Suppose that, on average, 1 person in 1000
makes a numerical error in preparing his or her income
tax return. If 10,000 returns are selected at random
and examined, find the probability that 6, 7, or 8 of
them contain an error.

Answer 1

We have n=10000 is large and p=1⁄1000 is near 0, then the binomial distribution can be approximated by the Poisson distribution with parameter
λ=np=10000×(1⁄1000)=10.
The probability that 6,7, or 8 of them contain an error
P(X=6)+P(X=7)+P(X=8)
Use Poisson distribution
P(X=x)=(e^(-λ) λ^x )/x! x=0,1,2,…
P(X=6)+P(X=7)+P(X=8)=(e^(-10) 〖10〗^6 )/6!+(e^(-10) 〖10〗^7 )/7!+(e^(-10) 〖10〗^8 )/8!=
=e^(-10) (〖10〗^6/40320)(56+80+100)≈0.265733715
Answer: 0.2657.