Fundamentals of Probability
Probability is the branch of mathematics that deals with the likelihood that certain outcomes will occur. There are five basic rules, or axioms, that one must understand while studying the fundamentals of probability.
Learning Objective

Explain the most basic and most important rules in determining the probability of an event
Key Points
 Probability is a number that can be assigned to outcomes and events. It always is greater than or equal to zero, and less than or equal to one.
 The sum of the probabilities of all outcomes must equal
$1$ .  If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
 The probability that an event does not occur is
$1$ minus the probability that the event does occur.  Two events
$A$ and$B$ are independent if knowing that one occurs does not change the probability that the other occurs.
Terms

experiment
Something that is done that produces measurable results, called outcomes.

outcome
One of the individual results that can occur in an experiment.

event
A subset of the sample space.

sample space
The set of all outcomes of an experiment.
Full Text
In discrete probability, we assume a welldefined experiment, such as flipping a coin or rolling a die. Each individual result which could occur is called an outcome. The set of all outcomes is called the sample space, and any subset of the sample space is called an event.
For example, consider the experiment of flipping a coin two times. There are four individual outcomes, namely
In probability theory, the probability
Probability Rules

Probability is a number. It is always greater than or equal to zero, and less than or equal to one. This can be written as
$0 \leq P(A) \leq 1$ . An impossible event, or an event that never occurs, has a probability of$0$ . An event that always occurs has a probability of$1$ . An event with a probability of$0.5$ will occur half of the time. 
The sum of the probabilities of all possibilities must equal
$1$ . Some outcome must occur on every trial, and the sum of all probabilities is 100%, or in this case,$1$ . This can be written as$P(S) = 1$ , where$S$ represents the entire sample space. 
If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. If one event occurs in
$30\%$ of the trials, a different event occurs in$20\%$ of the trials, and the two cannot occur together (if they are disjoint), then the probability that one or the other occurs is$30\% + 20\% = 50\%$ . This is sometimes referred to as the addition rule, and can be simplified with the following:$P(A \ \text{or} \ B) = P(A)+P(B)$ . The word "or" means the same thing in mathematics as the union, which uses the following symbol:$\cup $ . Thus when$A$ and$B$ are disjoint, we have$P(A \cup B) = P(A)+P(B)$ . 
The probability that an event does not occur is
$1$ minus the probability that the event does occur. If an event occurs in$60\%$ of all trials, it fails to occur in the other$40\%$ , because$100\%  60\% = 40\%$ . The probability that an event occurs and the probability that it does not occur always add up to$100\%$ , or$1$ . These events are called complementary events, and this rule is sometimes called the complement rule. It can be simplified with$P(A^c) = 1P(A)$ , where$A^c$ is the complement of$A$ . 
Two events
$A$ and$B$ are independent if knowing that one occurs does not change the probability that the other occurs. This is often called the multiplication rule. If$A$ and$B$ are independent, then$P(A \ \text{and} \ B) = P(A)P(B)$ . The word "and" in mathematics means the same thing in mathematics as the intersection, which uses the following symbol:$\cap$ . Therefore when A and B are independent, we have$P(A \cap B) = P(A)P(B).$
Extension of the Example
Elaborating on our example above of flipping two coins, assign the probability
1. Note that each probability is
2. Note that the sum of all the probabilities is
3. Suppose
4. The probability that no heads occurs is
5. If
Die
Dice are often used when learning the rules of probability.
Key Term Reference
 disjoint
 Appears in these related concepts: Unions and Intersections, Probability distributions, and The Poisson Random Variable
 independent
 Appears in these related concepts: Conditional Probability, Party Identification, and Probability Histograms
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, Averages, and Understanding Statistics
 multiplication rule
 Appears in these related concepts: The Multiplication Rule, The Collins Case, and Independence
 probability
 Appears in these related concepts: Theoretical Probability, Rules of Probability for Mendelian Inheritance, and The Addition Rule
 probability theory
 Appears in these related concepts: Applications of Statistics, Independence, and Complementary Events
 sample
 Appears in these related concepts: Identifying Product Benefits, Surveys, and Basic Inferential Statistics
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