## Basic Probability Definition

What is probability theory? A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

## Fundamental Terms

**Random Experiment** = A random experiment is a physical situation whose outcome cannot be predicted until it is observed.

**Sample Space** = A sample space, is a set of all possible outcomes of a random experiment.

**Random Variables** = A random variable, is a variable whose possible values are numerical outcomes of a random experiment. There are two types of random variables.

**Discrete Random Variable** is one which may take on only a countable number of distinct values such as 0,1,2,3,4,…….. Discrete random variables are usually (but not necessarily) counts.

2.

**Continuous Random Variable** is one which takes an infinite number of possible values. Continuous random variables are usually measurements.

**Probability** = Probability is the measure of the likelihood that an event will occur in a Random Experiment. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%).

**Conditional Probability** = Conditional Probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, is usually written as P(A|B).These are the most fundamental terms according to probability. Now let's have a look at some Formulas.

Basic Probability Formulas

#### Probability Range

0 ≤
*P*(A) ≤ 1

#### Rule of Complementary Events

*P*(A^{C})
+ *P*(A)
= 1

#### Rule of Addition

*P*(A∪B)
= *P*(A) + *P*(B) - *P*(A∩B)

#### Disjoint Events

Events A and B are disjoint iff

*P*(A∩B) = 0

#### Conditional Probability

*P*(A | B) = *P*(A∩B) / *P*(B)

#### Bayes Formula

*P*(A | B) = *P*(B | A) ⋅* P*(A)
/ *P*(B)

#### Independent Events

Events A and B are independent iff

*P*(A∩B) = *P*(A) ⋅ *P*(B)

#### Cumulative Distribution Function

*F _{X}*(

*x*) =

*P*(

*X*≤

*x*)

#### Probability Mass Function

_{
}

#### Probability Density Function

_{
}

_{
}

_{
}

_{
}

#### Covariance

#### Correlation

Bernoulli: 0-failure 1-success

Geometric: 0-failure 1-success

Hypergeometric: N objects with K success objects, n objects are taken.

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