Question

A telephone company's goal is to have no more than 4 monthly line failures on any 100 miles of line. The company currently experiences an average of 5 monthly line failures per 50 miles of line. Let x denote the number of monthly line failures per 100 miles of line. Assuming x has a Poisson distribution:


(a) Find the probability that the company will meet its goal on a particular 100 miles of line. (Do not round intermediate calculations. Round final answer to 4 decimal places.)


Probability

(b) Find the probability that the company will not meet its goal on a particular 100 miles of line. (Do not round intermediate calculations. Round final answer to 4 decimal places.)


Probability

(c) Find the probability that the company will have no more than 4 monthly failures on a particular 200 miles of line. (Do not round intermediate calculations. Round final answer to 4 decimal places.)


Probability

(d) Find the probability that the company will have more than 12 monthly failures on a particular 150 miles of line. (Do not round intermediate calculations. Round final answer to 4 decimal places.)

Answer 1

X : Number of monthly line failures per 100 miles of line .

Given ,  $X\sim Poi(10)$

The probability mass function of X is as follows :

$f(x)=P(X=x)=\frac{e^{-10}(10)^x}{x!}\ , x>0$

a )  $P(X\leq 4)=\sum_{x=0}^{4}P(X=x)= 0.02923$

b )  $P(X>4)=1-P(X\leq 4)=1- 0.0293=0.9707$

c ) Define , Y : Number of monthly line failures per 200 miles of line

$Y\sim Poi(20)$

$P(Y\leq 4)=0.0114$

d ) Define , M : Number of monthly line failures per 150 miles of line

$\therefore \ M\sim Poi(15)$

$P(M>12)=1-P(M\leq 12)=0.7324$