4. John needs to study potholes for his civil engineering project. He estimates

Question

4. John needs to study potholes for his civil engineering project. He estimates that a particular small road has an average of 1 pothole per mile and the potholes are located according to Poisson distribution. a) What is the length of the road he has to be ready to bike to be almost sure (specifically, 90% sure) he'll find at least three potholes? b) The weather today is bad, so John has decided he won't be able to bike more than 3 miles one way. He got lucky however and found two potholes after just one mile worth of biking. Given this information what is the probability of him finding at least three potholes today.

Explanation / Answer

small road has an average one pothole per mile and potholes are located according to poisson distribution.

poisson disribution haa mean lambda, from the given example lambda=1

x: no of pothole

a) to find length of the road that he will find the three potholes here x=3

therefore x*lambda=3*1= 3 miles length of the road that he will find the three potholes.

b) to find probability that john finding atleast 3 potholes today

P[x>=3]= 1-P(X<3) = 1-P(x=0)+P(x=1)+P(x=2)

pmf for poisson distribution

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

P(X=0)=e^-1*1^0/0! = 0.3678

P(X=1)=e^-1*1^1/1! = 0.3678

P(X=2)=e^-1*1^2/2!= 0.1839

P(X>3)= 1-[0.3678+0.3678+0.1839] =1-0.9195=0.0805

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