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.)
I would be thankful to get the solution.
To solve these problems, we can use the Poisson probability formula. The formula to calculate the probability of x events occurring in a given time period is:
P(x, λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x, λ) is the probability of x events occurring
- λ is the average number of events in the given time period
- e is Euler's number, approximately 2.71828
Let's solve each part of the problem:
(a) Find the probability that the company will meet its goal on a particular 100 miles of line.
In this case, λ (averge number of failures per 100 miles) is given as x. Therefore, λ = x. We are given that the goal is to have no more than 4 monthly failures, so x ≤ 4.
We need to calculate P(x ≤ 4, x). We can calculate this by summing the individual probabilities for x = 0, 1, 2, 3, and 4:
P(x ≤ 4, x) = P(x=0, x) + P(x=1, x) + P(x=2, x) + P(x=3, x) + P(x=4, x)
(b) Find the probability that the company will not meet its goal on a particular 100 miles of line.
To find the probability that the company will not meet its goal, we need to calculate the complement probability of part (a).
P(not meeting goal) = 1 - P(meeting goal)
(c) Find the probability that the company will have no more than 4 monthly failures on a particular 200 miles of line.
In this case, λ is the average number of failures per 200 miles. Since we are given the average number of failures for 50 miles, we need to multiply it by 4 to get the average number of failures for 200 miles. Therefore, λ = 4 * x.
We need to calculate P(x ≤ 4, 4 * x) similar to part (a), but with λ = 4 * x.
(d) Find the probability that the company will have more than 12 monthly failures on a particular 150 miles of line.
In this case, λ is the average number of failures per 150 miles. Since we are given the average number of failures for 50 miles, we need to multiply it by 3 to get the average number of failures for 150 miles. Therefore, λ = 3 * x.
We need to calculate P(x > 12, 3 * x). To calculate this, we can find the complement probability of P(x ≤ 12, 3 * x):
P(x > 12, 3 * x) = 1 - P(x ≤ 12, 3 * x)
By substituting the appropriate values of λ in the formula and calculating the probabilities using a calculator or software, we can find the answers to each part of the problem.