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.