Step 1 :The problem is asking for the probability that a randomly selected board cut by the machine has a length greater than 86.13 inches. This is a problem of normal distribution.
Step 2 :We can use the z-score formula to calculate the z-score, which is \((X - μ) / σ\), where X is the value we are interested in, μ is the mean, and σ is the standard deviation.
Step 3 :Substitute the given values into the formula: mean = 86, standard deviation = 0.3, and X = 86.13.
Step 4 :Calculate the z-score: \(z = (86.13 - 86) / 0.3 = 0.4333333333333182\).
Step 5 :After we get the z-score, we can use the standard normal distribution table or a function to get the probability. However, the table or function usually gives the probability that a value is less than X, so we need to subtract the result from 1 to get the probability that a value is greater than X.
Step 6 :Calculate the probability: \(P = 1 - 0.3323863126266806 = 0.6676\).
Step 7 :Final Answer: The probability that a randomly selected board cut by the machine has a length greater than 86.13 inches is approximately \(\boxed{0.6676}\).