# Supplemental Modules (Probability)

- Page ID
- 215

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

Probability theory is concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion.

- Area Under the Normal Curve and the Binomial Distribution
- Typically the probability distribution does not follow the standard normal distribution, but does follow a general normal distribution. When this is the case, we compute the z-score first to convert it into a standard normal distribution

- Probability and Independence
- For an experiment we define an event to be any collection of possible outcomes. A simple event is an event that consists of exactly one outcome. "or" means the union (i.e. either can occur) "and" means intersection (i.e. both must occur)

- Probability Distributions
- A variable whose value depends upon a chance experiment is called a random variable. Suppose that a person is asked who that person is closest to: their mother or their father. The random variable of this experiment is the boolean variable whose possibilities are {Mother, Father}. A continuous random variable is a variable whose possible outcomes are part of a continuous data set.

- The Central Limit Theorem
- Consider the distribution of rolling a die, which is uniform (flat) between 1 and 6. We will roll five dice we can compute the pdf of the mean. We will see that the distribution becomes more like a normal distribution. That is due to the Central Limit Theorem.

- The Normal Distribution and Control Charts
- The Random Distribution is a special distribution that we will use often and is a distribution for a continuous random variable.

- The z-score
- The number of standard deviations from the mean is called the z-score.

- Tree Diagrams and Counting
- A tree diagram is a diagram that branches out and ends in leaves that correspond to the final variety.