Question

Suppose that a busy traffic junction ,the probability p of an individual car having an accident is 0.0001. What is the probability of two or more cars being involved in an accident within this period?

Answer 1

Let n be the number of cars that pass through this traffic. Define a lambda for the Poisson distribution:

$\lambda=np=0.0001n$

For the Poisson distribution, the probability of k events is equal to

$P(k)= \frac{\lambda^k}{k!}e^{-\lambda}$

Then:

$P(k>2)=1-P(0)-P(1)\\ P(k>2)=1-e^{-\lambda}-\lambda e^{-\lambda}=1-e^{-0.0001n}(1+0.0001n)$

Due to the traffic load n tends to infinity:

$P(k>2)=\lim\limits_{n\to\infty}1-e^{-0.0001n}(1+0.0001n)$

The exponential function grows stronger than the power function:

$P(k>2)=1$