A factory produces glass for windows. The thickness X of an arbitrarily selected pane of glass is assumed to be Normally distributed with expectation μ = 4.10 and standard deviation σ = 0.04. Expectation and Standard deviation is measured in millimeters. What is the probability that an arbitrary route has a thickness less than 4.00 mm?

To find the probability that the thickness of a randomly selected pane of glass is less than 4.00 mm, we can use the standard normal distribution.

Step 1: Convert the given values to z-scores using the formula:

z = \frac{x - \mu}{\sigma}

where:
x is the value we want to find the probability for (in this case, 4.00 mm)
\mu is the mean of the distribution (4.10 mm)
\sigma is the standard deviation of the distribution (0.04 mm)

Plugging in the values, we get:

z = \frac{4.00 - 4.10}{0.04} = -2.5

Step 2: Look up the z-score in the standard normal distribution table or use a calculator to find the corresponding area under the curve.

Step 3: The area under the curve to the left of a z-score of -2.5 is the probability that a randomly selected pane of glass has a thickness less than 4.00 mm.

Using a standard normal distribution table, we find that the probability is approximately 0.0062.

Answer: The probability that an arbitrary pane of glass has a thickness less than 4.00 mm is approximately 0.0062.