Question

5. The occurrence of traffic accidents at an intersection is modeled as a Poisson process. Based on accident records at that intersection, a traffic engineer has determined that the average rate of accidents occurring at that intersection is once every three years.
(a) What is the probability that no accident occurs at that intersection for a period of 5 years?
(b) What is the probability that there is a traffic fatality at that intersection over a period of 3 years if for every accident there is 5% probability of fatality?

Answer 1

Answer 5. The occurrence of traffic accidents at an intersection is modeled as a Poisson process. Based on accident records at that intersection, a traffic engineer has determined that the average rate of accidents occurring at that intersection is once every three years.

Solution:

The average is 1 accident in 3 years.

Therefore, = 1/3

a) the probability that no accident occurs at that intersection for a period of 5 years:

Now, in 5 years the rate of accident will be

= (1/3)*5

= 1.67

the probability of no accident in 5 years:

P(X=0) = e^- * ^x/X!

= e^-1.67 * 1.67^0 / 0!

= 0.1882 * 1

= 0.1882

Therefore, the probability of no accident in 5 years is 0.1882.

b) the probability that there is a traffic fatality at that intersection over a period of 3 years if for every accident there is 5% probability of fatality:

there is 5% probability of fatality = 0.05

The rate of accident in 3 years will be

= (3/3) * 0.05

= 0.05

Therefore, the probability of accident in 3 years:

= 1 - P(X=0)

= 1 - e^- * ^x/X!

= 1 - e^-0.05 * 0.05^0 / 0!

= 1 - 0.9512

= 0.0488

Therefore, the probability of accident in 3 years is 0.0488.