A telephone company’s goal is to have no more than five monthly line failures on any 100 kilometres of line. The company currently experiences an average of two monthly line failures per 50 kilometres of line. Let x denote the number of monthly line failures per 100 kilometres of line. Assuming x has a Poisson distribution:
a) Find the probability that the company will meet its goal on a particular 100 kilometres of line.
b) Find the probability that the company will not meet its goal on a particular 100 kilometres of line.
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To find the probability that the company will meet its goal of no more than five monthly line failures on a particular 100 kilometers of line, we need to use the Poisson distribution.
a) In a Poisson distribution, the average rate (lambda) is given, which is the number of events (monthly line failures) per unit of measurement (50 kilometers in this case). The average rate lambda is calculated as follows:
lambda = (number of events) / (unit of measurement)
In this case, the average rate lambda is 2 monthly line failures per 50 kilometers. So lambda = 2.
Now, we need to find the probability of having no more than five monthly line failures on a particular 100 kilometers of line. We can use the Poisson probability formula:
P(X <= k) = (e^(-lambda) * lambda^k) / k!
Where P(X <= k) is the probability of having no more than k events (monthly line failures), lambda is the average rate, and k! is the factorial of k.
For k = 5, the probability will be:
P(X <= 5) = (e^(-2) * 2^5) / 5!
Substituting the values into the formula:
P(X <= 5) = (e^(-2) * 2^5) / (5 * 4 * 3 * 2 * 1)
Calculating this, we get:
P(X <= 5) ≈ 0.9837
Therefore, the probability that the company will meet its goal on a particular 100 kilometers of line is approximately 0.9837.
b) To find the probability that the company will not meet its goal on a particular 100 kilometers of line, we subtract the probability of meeting the goal from 1.
P(X > 5) = 1 - P(X <= 5)
Substituting the value for P(X <= 5) we found earlier:
P(X > 5) = 1 - 0.9837
Calculating this, we get:
P(X > 5) ≈ 0.0163
Therefore, the probability that the company will not meet its goal on a particular 100 kilometers of line is approximately 0.0163.