Probability Basics

Study Notes

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.

Description

Understand the fundamental concepts of probability, including experiments, outcomes, and sample spaces. Learn about different types of events, such as simple, compound, mutually exclusive, and independent events.