Question

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate alpha= 14 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.
(a) What is the probability that exactly ten arrive during the hour and all ten have no violations?

Answer 1

This question has two parts:

1. probability of 10 arriving in one hour, using Poisson distribution
\(P(X=k)=\dfrac{e^{\lambda} \lambda ^k}{k!}\)
where k=10, \(\lambda=14\), and

2. probability of all 10 arriving have no defects can be modelled using Binomial distribution,
\(P(X=r)=\large (^n_r)p^r (1-p)^{n-r}\)
where n=10 cars, r=10 without defects, p=probability of no defects

The answer is the product of the two.