Answer to Question #315697 in Statistics and Probability for Mirylle Anne

Question

Potholes on a highway can be a serious problem, and are in constant need of repair.

With a particular type of terrain and make of concrete, past experience suggests that

there are, on the average, 2 potholes per mile after a certain amount of usage. It is

assumed that the Poisson process applies to the random variable “number of

potholes.”

a) What is the probability that no more than one pothole will appear in a section of

1 mile?

b) What is the probability that no more than 4 potholes will occur in a given section

of 5 miles?

Answer 1

$a:\\X_1\sim Poiss\left( 2 \right) \\P\left( X_1\leqslant 1 \right) =P\left( X_1=0 \right) +P\left( X_1=1 \right) =\frac{2^0e^{-2}}{0!}+\frac{2^1e^{-2}}{1!}=0.406006\\b:\\X_5\sim Poiss\left( 10 \right) \\P\left( X_1\leqslant 4 \right) =P\left( X_1=0 \right) +P\left( X_1=1 \right) +P\left( X_1=2 \right) +P\left( X_1=3 \right) +P\left( X_1=4 \right) =\\=\frac{10^0e^{-10}}{0!}+\frac{10^1e^{-10}}{1!}+\frac{10^2e^{-10}}{2!}+\frac{10^3e^{-10}}{3!}+\frac{10^4e^{-10}}{4!}=0.0292527$