Probability Theory Fundamentals

Study Notes

Probability Theory

Basic Concepts

Experiment: An action or situation that can produce a set of outcomes
Sample Space: The set of all possible outcomes of an experiment
Event: A subset of the sample space
Probability: A numerical value between 0 and 1 that represents the likelihood of an event occurring

Probability Rules

Rule of Addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Rule of Multiplication: P(A ∩ B) = P(A) * P(B) if A and B are independent
Complement Rule: P(A') = 1 - P(A)

Types of Probability

Theoretical Probability: Based on the number of favorable outcomes divided by the total number of possible outcomes
Experimental Probability: Based on the number of times an event occurs in repeated trials
Subjective Probability: A personal estimate of the likelihood of an event

Conditional Probability

Conditional Probability: The probability of an event occurring given that another event has occurred
Formula: P(A|B) = P(A ∩ B) / P(B)
Independence: Two events are independent if P(A|B) = P(A)

Bayes' Theorem

Formula: P(A|B) = P(B|A) * P(A) / P(B)
Used to update: The probability of an event based on new information

Random Variables

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

Description

Test your understanding of basic probability concepts, including experiments, sample spaces, events, and probability rules. This quiz covers theoretical, experimental, and subjective probability, as well as conditional probability and Bayes' Theorem.