# Fundamentals of Probability in a Finite Sample Space

• A sample space is a set of all possible outcomes of an experiment.
• An event is a subset of the sample space.
• The probability of an event A, denoted by P(A), is the ratio of the number of favourable outcomes to the number of possible outcomes.
• 0 ≤ P(A) ≤ 1. Probabilities are always numbers between 0 (impossible) and 1 (certain).
• Probabilities across all outcomes in a sample space always sum to 1.

# Probabilities of Combined Events

• The Union of events A and B represents the event that either A, or B, or both occur.
• P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• The Intersection of events A and B represents the event that both A and B occur.
• When events A and B are independent (the outcome of A does not affect the outcome of B), P(A ∩ B) = P(A) × P(B).
• Events A and B are mutually exclusive if they can’t both occur at the same time. For mutually exclusive events, P(A ∩ B) = 0.

# Conditional Probability

•  The conditional probability of an event A given that another event B has occurred is denoted by P(A B).
•  **P(A B) = P(A ∩ B) / P(B)**, provided that P(B) ≠ 0.

# Use of Probability Trees

• Probability trees can be useful for calculating probabilities in complex scenarios.
• Each branch of the tree represents a possible outcome.
• The product of probabilities along a path provides the probability of that set of outcomes.

# Bayes’ Theorem

• Bayes’ Theorem can be used to update probabilities based on new information.
•  The theorem states: **P(A B) = [ P(A) × P(B A) ] / P(B)**

Remember to practice using a variety of problems to gain experience in combining these principles.