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The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean, and about 95% falls within two standard deviations. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. To find probabilities for any normal distribution, we can convert the values to z-scores using the formula z = (x - μ) / σ. This transformation allows us to use standard normal distribution tables or software to find probabilities associated with any normal distribution.

To calculate the probability of a random variable falling within a specific range in a normal distribution, we find the z-scores for the endpoints of the range and then use the standard normal distribution to determine the area under the curve between these z-scores. This area represents the probability that the random variable falls within the specified range, which can be found using cumulative distribution functions or z-tables.