Probability Basics

Definition of Probability

• Probability is a measure of the likelihood of an event occurring
• It is a number between 0 and 1, where:
  • 0 represents an impossible event
  • 1 represents a certain event
  • Values between 0 and 1 represent the degree of uncertainty

Basic Concepts

• Experiment: an action or situation that can produce a set of outcomes
• Outcome: a specific result of an experiment
• Sample Space: the set of all possible outcomes of an experiment
• Event: a set of one or more outcomes of an experiment

Rules of Probability

• Probability of an Event: the sum of the probabilities of its outcomes
• Probability of the Union of Events: the sum of the probabilities of each event, minus the probability of their intersection
• Probability of the Intersection of Events: the product of the probabilities of each event, assuming independence
• Complement of an Event: the set of outcomes not in the event, with probability 1 - P(event)

Types of Events

• Independent Events: the occurrence of one event does not affect the probability of the other
• Dependent Events: the occurrence of one event affects the probability of the other
• Mutually Exclusive Events: the occurrence of one event means the other cannot occur
• Exhaustive Events: the events cover all possible outcomes of the sample space

Conditional Probability

• Conditional Probability Formula: P(A|B) = P(A ∩ B) / P(B)
• Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)

Random Variables

• Discrete Random Variable: takes on a countable number of distinct values
• Continuous Random Variable: takes on an uncountable number of values in an interval
• Probability Distribution: a function describing the probability of each value of a random variable

Definition of Probability

• Probability is a measure of the likelihood of an event occurring, represented by a number between 0 and 1
• 0 represents an impossible event, 1 represents a certain event, and values between 0 and 1 represent the degree of uncertainty

Basic Concepts

• An Experiment is an action or situation that can produce a set of outcomes
• An Outcome is a specific result of an experiment
• The Sample Space is the set of all possible outcomes of an experiment
• An Event is a set of one or more outcomes of an experiment

Rules of Probability

• The Probability of an Event is the sum of the probabilities of its outcomes
• The Probability of the Union of Events is the sum of the probabilities of each event, minus the probability of their intersection
• The Probability of the Intersection of Events is the product of the probabilities of each event, assuming independence
• The Complement of an Event is the set of outcomes not in the event, with probability 1 - P(event)

Types of Events

• Independent Events: the occurrence of one event does not affect the probability of the other
• Dependent Events: the occurrence of one event affects the probability of the other
• Mutually Exclusive Events: the occurrence of one event means the other cannot occur
• Exhaustive Events: the events cover all possible outcomes of the sample space

Conditional Probability

• The Conditional Probability Formula is P(A|B) = P(A ∩ B) / P(B)
• Bayes' Theorem is P(A|B) = P(B|A) * P(A) / P(B)

Random Variables

• A Discrete Random Variable takes on a countable number of distinct values
• A Continuous Random Variable takes on an uncountable number of values in an interval
• A Probability Distribution is a function describing the probability of each value of a random variable