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Probability theory is the mathematical framework for quantifying uncertainty. It involves predicting the likelihood of different outcomes based on a set of possible events. In probability theory, we use

• Events: defined outcomes or occurrences.
• Sample Space: the set of all possible outcomes.
• Probability Measures: numerical values assigned to the likelihood of events.

Probabilities are expressed as numbers between 0 and 1, where 0 means an event is impossible, and 1 means it is certain.
The theory has broad applications across various fields such as finance, science, and engineering, where we evaluate uncertainties and make predictions. Learning the foundational concepts of probability theory is crucial, especially when moving on to more complex topics like continuous probability distributions.

The normal distribution is a key concept in statistics and probability theory, often known as the bell curve due to its shape. It represents a type of continuous probability distribution for a random variable and

• has a symmetric shape about the mean.
• characterizes many natural phenomena, such as human heights and test scores.
• is defined by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).

An important property of the normal distribution is that it is completely described by its mean and standard deviation.
The mean locates the center of the distribution, while the standard deviation provides a measure of the spread of the distribution. The data within one standard deviation from the mean accounts for about 68% of the measurements, within two standard deviations for about 95%, and within three standard deviations for about 99.7%.
Understanding the normal distribution is essential for many statistical analyses, which often assume data to be normally distributed.

Probability intervals are a fundamental concept when dealing with continuous probability distributions. These intervals refer to the range or interval in which a continuous variable falls and

• represent how probabilities are calculated over a continuum.
• can be denoted using inequalities involving variables.
• are not affected by whether they include or exclude their endpoints in continuous distributions.

For example, in a normal distribution, saying that a variable falls within the interval \((a, b)\) is the same as saying \((a, b]\), \([a, b)\), or \([a, b]\), since the probability of a precise point in a continuous interval is zero.
This is fundamentally different from discrete probability distributions, where the inclusion or exclusion of interval endpoints matters.
Understanding probability intervals helps in grasping why statements such as \(P(a