Probability of Events in a Finite Sample Space
Probability of Events in a Finite Sample Space
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.