Probability Theory Fundamentals

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