# Probability Index

Probability is used to estimate the likely-hood of an event. Probabilities are always non-negative and sum to one.

## Notation

In probability a random variable is denoted using capital latin letters usually X, Y, and Z or A, and B respectively.

A probability distribution is denoted like a function. Often a capital P is used for the function name, and the capital letter X is used for the argument to the function.

Conditional probability is denoted using a vertical bar between the two variables

A joint probability distribution is denoted like a function often using P as the function name. The capital letters X and Y are often used to represent the random variables of the distribution and are arguments to the function.

The expected value of a probability distribution is denoted as the function E(X).

The arithmetic mean of a data set is denoted by a horizontal bar over the variable x.

The number of possible ways to choose r combinations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.

The number of possible ways to choose r permutations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.

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## Formulas

The combination formula describes the possible combinations of r elements out of a group of n elements where order does not matter.

The permutation formula describes the possible permutations of r elements out of a group of n elements where order does matter.

The number of permutations of n distinct items is given by n factorial. A permutation is a unique ordering or arangment of the set of items.

The arithmetic mean, also called the sample mean, is the average of a sample space. To calculate the arithmetic mean sum all the data points in a sample space and then divide by the number of elements.

The conditional probility formula shows how to calculate the probability of a event B, given that another event A has already occured.

The expected value, describes the most likely value of a probability distribution. It also describes where a probability distribution is centered.

To calculate the expected value of a discrete distribution multiply all of the events of the distribution with the probability of the element occuring.

The sample standard deviation formula is denoted by the greek lower case sigma symbol in the case of the population and the latin letter s for the sample.

## Concepts

A random variable represents an event whose outcome is unknown.

Bayes theorem describes a way of expressing conditional probability.

## Examples

To calculate n choose r where order does not matter you can use the formula for four choose two combinations.

To calculate n choose r where order matters you can use the permuation formula.

To calculate the possible permutations of n distinct items, you can take the factorial of n to get the numer of permutations.

The population mean, sometimes called the expected value, describes where a probability distribution is centered.

The population mean, sometimes called the expected value, describes where a probability distribution is centered.

Variance measures how spread out a distribution is.