Question

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Answer 1

Solution:

(a) Find the probability that the company will meet the goal on a particular 100 miles of line?

Answer: Let X denotes the number of monthly line failures per 100 miles of line. Then we have:

$\small \lambda=\frac{6}{50} \times 100=12$

Therefore, we have to find here:

$\small P(X\leq 4)$

Using the Poisson distribution, we have:

$\small P(X\leq 4)=0.0076$

$\small \boldsymbol{\therefore Probability=0.0076 }$

(b) Find the probability that the company will not meet the goal on a particular 100 miles of line?

Answer: $\small \boldsymbol{Probability =P(X>4)=1-P(X\leq 4)=1-0.0076=0.9924}$

(c) Find the probability that the company will have no more than 4 monthly failures on a particular 200 miles of line?

Answer: Let X denotes the number of monthly line failures per 200 miles of line. Then we have:

$\small \lambda=\frac{6}{50} \times 200=24$

Therefore, we have to find here:

$\small P(X\leq 4)$

Using the Poisson distribution, we have:

$\small P(X\leq 4)=0.0000$

$\small \boldsymbol{\therefore Probability =0.0000 }$

(d) Find the probability that the company will have more than 12 monthly failures on a particular 150 miles of line?

Answer: Let X denotes the number of monthly line failures per 150 miles of line. Then we have:

$\small \lambda=\frac{6}{50} \times 150=18$

Therefore, we have to find here:

$\small P(X> 12)$

Using the Poisson distribution, we have:

$\small P(X> 12)=1-P(X\leq 12)=1-0.0917=0.9083$

$\small \boldsymbol{\therefore Probability =0.9083}$