Probability Overview

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  1. Theories of Probability
    1. Introduction
    2. Classical Theory
    3. Relative Frequency Theory </aside>

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  1. Axioms
    1. Addition Rule
    2. Multiplication Rule
    3. Rule of At Least One Success
    4. Expected Number of Trials
    5. Bayes’ Theorem </aside>

Theories of Probability

  1. Introduction

    1. Probability is the branch of mathematics that quantifies the uncertainty of events. Experiments conducted under homogeneous and similar conditions can result in two types of outcomes:
    2. Unique or Certain Outcomes
      1. Also known as deterministic phenomena.
      2. These are predictable results that remain constant when the experiment is repeated under the same conditions.
      3. Example: The Sun rises in the East and sets in the West.
    3. Not Unique, but One of Several Possible Outcomes
      1. Known as probabilistic phenomena.
      2. These outcomes are unpredictable, with one outcome selected from a set of possibilities.
      3. Example: Tossing a coin results in either heads or tails.

  2. Key Terminologies in Probability

    1. Trial
      1. A trial is an experiment conducted under identical conditions that may result in one of several outcomes.
      2. Example: Tossing a coin is a trial where the possible outcomes are heads or tails.
    2. Events
      1. An event is an outcome or a set of outcomes of a trial.
      2. Example: Rolling a die and obtaining a 4 is an event.
    3. Exhaustive Events
      1. Definition: The total number of all possible outcomes in a trial.
      2. Example: For a six-sided die, the exhaustive events are {1, 2, 3, 4, 5, 6}.
    4. Favorable Events
      1. Definition: Outcomes that result in the occurrence of the event of interest.
      2. Example: Rolling an even number (2, 4, 6) on a die are favorable events for the event "rolling an even number."
    5. Mutually Exclusive Events
      1. Definition: A set of events such that the occurrence of one event excludes the occurrence of all others.
      2. Example: For a coin toss, the events "heads" and "tails" are mutually exclusive because they cannot occur simultaneously.
    6. Equally Likely Events
      1. Definition: Events for which, after considering all relevant evidence, there is no reason to expect one event over the other.
      2. Example: Rolling a die has equally likely events of {1, 2, 3, 4, 5, 6}.
    7. Independent Events
      1. Definition: A set of events where the occurrence or non-occurrence of one does not affect the occurrence of the others.
      2. Example: Tossing two separate coins. The result of the first toss does not influence the result of the second.

  3. Probability of an Event

    1. Marginal (Single/Unconditional) Probability

      1. Definition: The probability of a single event occurring, considering no other events.

      2. Represented as $\text{P(A)}$, where $\text{A}$ is the event of interest.

        image.png

      3. Example: Tossing a coin:

        $$ P(\text{Heads}) = \frac{1}{2} $$

        $$ P(\text{Tails}) = \frac{1}{2} $$

      4. Rolling a six-sided die:

        $$ P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \frac{1}{6} $$

      5. Called "marginal probability" because it is often listed in the margins of probability tables.

  4. Visualization Using Venn Diagrams

    1. Sample Space Representation

      1. The entire sample space is represented as a rectangle.

        image.png

      2. Each possible event is represented as a part of the rectangle.

    2. Two Mutually Exclusive Events
      1. If two events $A$ and $B$ are mutually exclusive:
      2. Diagram:
    3. Two Non-Mutually Exclusive Events

  5. Classical Theory of Probability

  6. Relative Theory of Probability