Question

Suppose that at a busy traffic junction, the probability p of an individual car having an accident is 0.0001. During a certain part of the day, 100 cars pass through the junction. What is the probability of two or more cars being involved in an accident within this period?

Answer 1

This is a binomial distribution with $n=100,\;p=0.0001.$

The binomial distribution can be approximated by the Poisson distribution with $\lambda=np=0.01.$

$P(X\ge2)=1-P(X=0)-P(X=1)=e^{-\lambda}(\frac{\lambda^0}{0!}+\frac{\lambda^1}{1!})=$

$=1-e^{-0.01}(1+0.01)=0.00004967.$