In a manufacturing facility, a specific machine produces cylindrical components, and

the diameter of these components follows a normal distribution with a mean diam$-$

eter of $50$ millimeters and a standard deviation of $2$ millimeters. Quality control

measures are in place to ensure that most components meet the required specifica$-$

tions.

$($a$)$ Calculate the probability that a randomly selected component has a diameter

between $48$ and $52$ millimeters.

$($b$)$ The company has strict quality standards, and components with diameters

less than $48$ millimeters or greater than $52$ millimeters are considered defects.

What is the probability of a randomly selected component being a defect?

$($c$)$ To optimize production, the company wants to produce components with a

diameter greater than $52$ millimeters. What percentage of components meet

this requirement?

$($d$)$ If the company wants to introduce a new quality control threshold at $51$ mil$-$

limeters$,$ what is the minimum percentage of components that should meet this

threshold to maintain a high level of quality $($i$.$e$.,$ less than $5\%$ defective$)?$