If a random variable is normal, you can map it to a standard normal distribution (useful for finding probabilities in the standard normal table) by the following relationship:
Example 1: is normal. and Then
Example 2: has the same distribution as example 1. Then implies
With regard to Central Limit Theorem:
By the Central Limit Theorem, the distribution of a sum of iid random variables converges to a normal distribution as the number of iid random variables increases. This means that if the number of iid random variables is sufficiently large, we can get approximate probabilities by using a normal distribution approximation.