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
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To solve this problem, we can use the Poisson distribution formula. The formula for the Poisson distribution is:
P(x; λ) = (e^-λ * λ^x) / x!
In this case, we want to find the probability that the company will have no more than 4 monthly line failures on 100 miles of line. The average number of monthly line failures per 100 miles of line is twice the average number of line failures per 50 miles, so λ = 2 * 5 = 10.
Let's calculate the probability:
P(x ≤ 4; λ) = P(x = 0; λ) + P(x = 1; λ) + P(x = 2; λ) + P(x = 3; λ) + P(x = 4; λ)
= (e^-10 * 10^0) / 0! + (e^-10 * 10^1) / 1! + (e^-10 * 10^2) / 2! + (e^-10 * 10^3) / 3! + (e^-10 * 10^4) / 4!
= (1 * 10^0) / 1 + (0.0000454 * 10^1) / 1 + (0.0000454 * 10^2) / 2 + (0.0000454 * 10^3) / 6 + (0.0000454 * 10^4) / 24
= 0.000454 + 0.000454 + 0.00454 + 0.01513 + 0.03782
= 0.05848
Therefore, the probability that the company will meet its goal on a particular 100 miles of line is approximately 0.05848 or 5.848%.
To find the probability that the company will meet its goal on a particular 100 miles of line, we need to calculate the probability of having no more than 4 monthly line failures on this section of the line.
Let's define the average number of monthly line failures per 100 miles of line as x. The given information states that the company experiences an average of 5 monthly line failures per 50 miles of line. Therefore, the average number of monthly line failures per 100 miles of line is 2 * 5 = 10.
Now, we can use the Poisson distribution formula to calculate the probability:
P(X ≤ 4) = e^(-λ) * (λ^0/0!) + e^(-λ) * (λ^1/1!) + e^(-λ) * (λ^2/2!) + e^(-λ) * (λ^3/3!) + e^(-λ) * (λ^4/4!)
Where λ is the average number of events per unit of time or space (in this case, the average number of monthly line failures per 100 miles of line).
In our case, λ = 10. Plugging in the values:
P(X ≤ 4) = e^(-10) * (10^0/0!) + e^(-10) * (10^1/1!) + e^(-10) * (10^2/2!) + e^(-10) * (10^3/3!) + e^(-10) * (10^4/4!)
Using a calculator or software that can evaluate the exponential function and factorial, we can find the exact value of P(X ≤ 4). The result will give us the probability that the company will meet its goal on a particular 100 miles of line.