Review of Basic Probability - Part 1

Objective:

• By the end of this lesson, students will have reviewed the following topics:

  • Definition of events
  • Definition of sample space
  • Definition of probability
  • Law of Large Numbers
  • Addition Rule for mutually exclusive events
  • General Addition Rule
  • Complement Rule

Duration:

• 75 minutes

Materials:

• Handouts with exercises and problems related to basic probability
• Computer, projector, and screen

Introduction:

• Examples that can be used to jump start topic:

  • birthday problem

  • Monty Hall problem

  • casino games

Introduce the lesson’s topic:

• Today we will review basic probability rules.

Historical Context:

• 1560s: Cardano wrote Liber de ludo aleae, the first known systematic treatment of probability, and as the result of a gambling addiction.

• 1654: Fermat and Pascal worked on the foundation of probability theory through correspondence.

• 1812 and 1814: Laplace published Théorie analytique des probabilités and Essai philosophique sur les probabilités, outlining many basic and fundamental results in statistics.

Main Content:

• Review of Events, Sample Spaces, and Probability:

  • Probability: a number between 0 and 1 that measures the uncertainty of a particular event.

    • \(p = 0 \to\) event will never happen.

    • \(p = 1 \to\) event will definitely happen.

  • Intuitive probability:

    • What is the probability of being pulled over while speeding in Gulf Breeze?

    • What is the probability of seeing a penguin walk around UWF?

    • What is the probability of seeing an armadillo walk around UWF?

  • Events: an outcome (or collection of outcomes) in a statistical experiment

    • Events can be defined and described in three ways (1.7 in Albert and Hu):

      1. \(A \cap B\) is the intersection between \(A\) and \(B\); it is the event that both \(A\) and \(B\) occur.

      2. \(A \cup B\) is the union between \(A\) and \(B\); it is the event that either \(A\) or \(B\) occur.

      3. \(A^c\) is the complement of \(A\); it is the event that \(A\) does not occur.

  • Sample spaces: all possible outcomes of a statistical experiment

• Addition Rules

  • Mutually exclusive events: events that have no outcomes in common; also known as disjoint

    • Venn Diagram examples

  • Addition Rule for Mutually Exclusive Events: if \(A\) and \(B\) are mutually exclusive events, then \[P(A \cup B) = P(A) + P(B)\]

    • Examples: 1 instructor works through, 2 students work through

  • General Addition Rule: regardless of \(A\) and \(B\) being mutually exclusive events, \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

    • Examples: 1 instructor works through, 2 students work through

  • Complements: if \(S\) is the sample space, then the complement of event \(A\), denoted by \(A^c\) is all outcomes in \(S\) that are not outcomes in event \(A\).

  • Complement Rule: consider any event \(A\) and its complement, \(A^c\). From probability rules, we know \[P(A^c) = 1 - P(A)\]

    • Examples: 1 the instructor works through, 2 students work through

Calculation and Practice:

• Examples for events and sample spaces:

  • Example 1:

  • Example 2:

  • Example 3:

• Examples for the Addition Rule for Mutually Exclusive Events:

  • Example 1

  • Example 2

  • Example 3

• Examples for the General Addition Rule:

  • Example 1

  • Example 2

  • Example 3

Discussion and Wrap-Up:

• Facilitate a class discussion to review the example problems, reinforce key concepts, and answer any questions the students have.

Homework:

• Assign additional problems to practice the basic probability rules.

Formative Assessment:

• Evaluate students based on their participation in discussions, their ability to solve example problems, and their performance on the assigned homework.

Conclusion:

• Emphasize these are building blocks for the next lesson and long term understanding probability.