A telephone company's goal is to have no more than four monthly line failures on any 50 miles of line. The company currently experiences an average of 4 monthly line failures per 25 miles of line. Let x denote the number of monthly line failures per 50 miles of line. Assuming x has a Poisson distribution
To solve this problem, we need to use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space.
The formula for the Poisson distribution is:
P(x; μ) = (e^(-μ) * μ^x) / x!
Where:
- P(x; μ) represents the probability of having exactly x events in the interval
- e is Euler's number (approximately 2.71828)
- μ is the expected number of events in the interval
- x is the actual number of events we are interested in
In this case, the company's goal is to have no more than 4 monthly line failures on any 50 miles of line. We are given that the company currently experiences an average of 4 monthly line failures per 25 miles of line. We want to find the probability of having no more than x line failures per 50 miles of line.
To solve this, let's first calculate the new average number of line failures in 50 miles. Since the current average is 4 failures per 25 miles, the average for 50 miles can be calculated as follows:
(4 failures / 25 miles) * 50 miles = 8 failures
So, the average number of line failures per 50 miles is 8.
Now, let's calculate the probability of having no more than 4 line failures per 50 miles using the Poisson distribution formula.
P(x ≤ 4; μ = 8) = P(0; 8) + P(1; 8) + P(2; 8) + P(3; 8) + P(4; 8)
Using the Poisson distribution formula, you can plug in the values:
P(0; 8) = (e^(-8) * 8^0) / 0! = (e^(-8)) / 1 ≈ 0.0003
P(1; 8) = (e^(-8) * 8^1) / 1! = (e^(-8) * 8) ≈ 0.0023
P(2; 8) = (e^(-8) * 8^2) / 2! = (e^(-8) * 64) ≈ 0.0092
P(3; 8) = (e^(-8) * 8^3) / 3! = (e^(-8) * 512) ≈ 0.0245
P(4; 8) = (e^(-8) * 8^4) / 4! = (e^(-8) * 4096) ≈ 0.049
Add them up to get the total probability:
P(x ≤ 4; μ = 8) ≈ 0.0003 + 0.0023 + 0.0092 + 0.0245 + 0.049 ≈ 0.0853
Therefore, the probability of having no more than 4 line failures per 50 miles is approximately 0.0853.