Probability Basics

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.

Types of Events

• Simple Event: An event that consists of a single outcome.
• Compound Event: An event that consists of multiple outcomes.
• Mutually Exclusive Events: Events that cannot occur at the same time.
• Independent Events: Events whose occurrence does not affect the probability of other events.

Probability Rules

• Probability Axioms:
  • The probability of any event is a number between 0 and 1.
  • The probability of the sample space is 1.
  • The probability of the empty set is 0.
• Addition Rule: The probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection.
• Multiplication Rule: The probability of the intersection of two independent events is the product of their individual probabilities.

Probability Measures

• Theoretical Probability: The probability of an event based on the number of favorable outcomes divided by the total number of possible outcomes.
• Experimental Probability: The probability of an event based on the results of repeated trials.

Conditional Probability

• Conditional Probability Formula: The probability of an event given that another event has occurred.
• Independent Events: Events with conditional probability equal to their unconditional probability.

Bayes' Theorem

• Bayes' Formula: A formula for updating the probability of an event based on new information.
• Applications: Medical diagnosis, spam filtering, and machine learning.

Random Variables

• Discrete Random Variable: A variable that can take on a countable number of distinct values.
• Continuous Random Variable: A variable that can take on any value within a certain range or interval.
• Probability Distribution: A function that describes the probability of each possible value of a random variable.

Basic Concepts

• An experiment is a situation that can produce a set of outcomes, and an outcome is a specific result of an experiment.
• The sample space is the set of all possible outcomes of an experiment.

Types of Events

• A simple event consists of a single outcome, while a compound event consists of multiple outcomes.
• Mutually exclusive events are events that cannot occur at the same time, and independent events are events whose occurrence does not affect the probability of other events.

Probability Rules

• The probability of any event is a number between 0 and 1, and the probability of the sample space is 1.
• The probability of the empty set is 0, and the probability of an event is equal to 1 - the probability of its complement.
• The addition rule states that the probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection.
• The multiplication rule states that the probability of the intersection of two independent events is the product of their individual probabilities.

Probability Measures

• Theoretical probability is the probability of an event based on the number of favorable outcomes divided by the total number of possible outcomes.
• Experimental probability is the probability of an event based on the results of repeated trials.

Conditional Probability

• The conditional probability of an event given that another event has occurred is calculated using the conditional probability formula.
• Independent events have conditional probability equal to their unconditional probability.

Bayes' Theorem

• Bayes' formula is a formula for updating the probability of an event based on new information.
• Bayes' theorem has applications in medical diagnosis, spam filtering, and machine learning.

Random Variables

• A discrete random variable can take on a countable number of distinct values, while a continuous random variable can take on any value within a certain range or interval.
• A probability distribution is a function that describes the probability of each possible value of a random variable.